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International Journal of Non-Linear Mechanics 85 (2016) 161173

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International Journal of Non-Linear Mechanics

http://d0020-74

n CorrE-m

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Buckling and nonlinear dynamics of elastically coupled double-beamsystems

Ivana Bochicchio a,n, Claudio Giorgi b, Elena Vuk b

a DipMat, Universit degli studi di Salerno, Italyb DICATAM, Universit degli studi di Brescia, Italy

a r t i c l e i n f o

Article history:Received 9 December 2015Received in revised form17 June 2016Accepted 26 June 2016Available online 29 June 2016

MSC:35B4135G5035Q7474B2074G6074H4074K10

Keywords:Double-beam systemSteady statesBucklingNonlinear oscillationsGlobal attractor

x.doi.org/10.1016/j.ijnonlinmec.2016.06.00962/& 2016 Elsevier Ltd. All rights reserved.

esponding author.ail address: [email protected] (I. Bochicchio

a b s t r a c t

This paper deals with damped transverse vibrations of elastically coupled double-beam system undereven compressive axial loading. Each beam is assumed to be elastic, extensible and supported at theends. The related stationary problem is proved to admit both unimodal (only one eigenfunction is in-volved) and bimodal (two eigenfunctions are involved) buckled solutions, and their number depends onstructural parameters and applied axial loads. The occurrence of a so complex structure of the steadystates motivates a global analysis of the longtime dynamics. In this regard, we are able to prove theexistence of a global regular attractor of solutions. When a finite set of stationary solutions occurs, itconsists of the unstable manifolds connecting them.

& 2016 Elsevier Ltd. All rights reserved.

1. Introduction

In this paper we investigate the properties of damped trans-verse vibrations and dynamical buckling of a coupled double-beam system under even compressive axial loading. The systemmodels a sandwich structure with an elastic filler. It is composedof two equal WK-beams (according to the nonlinear model ofWoinowsky-Krieger [37]), which are connected by linear springsand simply supported at the ends (see Fig. 1).

The in-plane dynamics is ruled by the following nonlinear system:

( )

( )

+ + +

+ [ ] =

+ + +

[ ] = ( )

( )

( )

u u u u u

u u f

u u u u u

u u f

,

, 1

tt xxxx t x L L xx

tt xxxx t x L L xx

1 1 1 1 0,2

1

1 2 1

2 2 2 2 0,2

2

1 2 2

2

2

).

where the unknown variables [ ] + u L: 0,i (i1,2) representthe downward deflections in the vertical plane of the midline of thebeams with respect to their reference configuration at rest. Bothbeams are hinged at their ends, so that

( )( ) = ( ) = ( ) = ( ) = [ ) = 2u t u L t u t u L t t i0, , 0, , 0, 0, , 1, 2.i i xx i xx i

The unknown fields are required to satisfy the following initial con-ditions:

( ) = ( ) ( ) = ( ) [ ] = ( )u x u x u x v x x L i, 0 , , 0 , 0, , 1, 2, 3i i t i i0 0

where u10, u2

0, v10 and v2

0 are given functions which fulfill (2). The WK-beams are assumed to have equal length L and unitary mass. In thereference (natural) configuration they are straight and parallel, andtheir spacing is d. They are connected by linear elastic springs withcommon stiffness and free length d. Sources fi, i1,2, represent thegiven vertical load distributions. The positive constants and denotethe flexural rigidity of the beams and the viscosity of the externalenvironment, respectively. Finally, is a positive constant, whereas theparameter summarizes the effect of the axial force acting at one endof each beam and is positive when both beams are stretched, negativewhen compressed.

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Fig. 1. In-plane oscillations of a double-beam sandwich system.

I. Bochicchio et al. / International Journal of Non-Linear Mechanics 85 (2016) 161173162

A derivation of the WK-beam model within a thermoelasticframework can be found in [18]. According to the modeling ap-proach devised therein, the WK-beam equation is a simplificationof the nonlinear von Krmn one-dimensional model, wherelongitudinal (horizontal in our case) displacements are condensedby integrating the corresponding equation in which longitudinalinertia is neglected. Unlike the usual EulerBernoulli linear theory,a nonlinear but uniform term accounting for extensibility of thebeam is retained into the equation of the transversal vibration. Westress that all material constants in (1) are dimensionless (see [18]for details). In particular

= > = hL

DL6

0,2

,2

2

where h and D are the thickness of the beam and its longitudinaldisplacement at the ends, respectively. As usual, both are assumedto be considerably shorter than the length L. Finally, we remarkthat the elongation of the coupling springs must take account ofthe horizontal displacements of their anchor points. Nevertheless,in (1) the strain in the springs is approached by the differencebetween the vertical displacements of the two beams. This may beaccounted for assuming the maximum horizontal displacement,| |D , to be negligible if compared with the reference spacing of thebeams, namely | |D d.

System (1) may be also used to describe out-of-plane oscilla-tions, both vertical and torsional, of a girder bridge where the roadbed is modeled by an elastic rug connecting two lateral WK-beams(see Fig. 2). In this case, however, the lateral movements of thebeams are neglected by the model.

Although all results obtained hereafter apply to both materialmodels, for the sake of definiteness we shall refer to the former,only. In addition, due to the recasting of the problem into an ab-stract setting, we stress that the present analysis can be easilyextended to (Berger) plate-type sandwich structures with hingedand normally loaded boundaries. In spite of a relatively wide lit-erature concerning statics and dynamics of a single WK-beam (seee.g. [2,3,5,7,8,14,15,19,22,31,37] and references therein), we arenot aware of analytic studies which consider the elastic coupling oftwo or more nonlinear beams of this type. On the other hand,mathematical models of sandwich beam-type and plate-typestructures raised a wide interest in the literature, due to their re-levance in many branches of modern civil, mechanical and aero-space engineering. In particular in the 80s the phenomenon ofnonlinear buckling mode interaction stimulated much interest andhas been investigated by many authors. In particular we recall thefundamental contribution by Budiansky [11] and the many tech-nical papers by Sridharan (see for instance [4,32]). Recently, after

Fig. 2. Out-of-plane oscillations of a double-beam girder bridge.

some pioneer works (see, for instance, [23,29]) concerning inter-action buckling between two beams, a lot of papers deal withmechanical properties of axially loaded and elastically connectedlinear double beam systems (see, for instance, [21,25,26,35,36,38,39]).

The aim of this paper is to give a contribution on this subject byscrutinizing statics and dynamics of the initial boundary valueproblem (1)(3). Its novelty and relevance relies on the completecharacterization of the long time behavior, which emphasizes thedifferent behavior of the nonlinear system (1) with respect todouble-beam linear systems previously considered. It is worthnoting that in a wider context concerning the localization of vi-bration modes and buckling patterns [24,27,30], nonlinearitiesplay the same role than imperfections do in linear systems.

The occurrence of a very complex structure of the steady statesmotivates a global analysis of the longtime dynamics of system (1)(Sections 4 and 5). In this regard, due to the dissipative nature ofthe system ( > 0), we are able to prove the existence of a globalregular attractor of solutions. In particular, when a finite set ofstationary solutions occurs, the global attractor is given by theunion of the unstable manifolds connecting them (Section 5.3).

2. Preliminary results

Introducing a suitable functional framework, we recast theoriginal system (1) into an abstract setting. Let ( )H, , , be areal Hilbert space, and let ( ) A A H H: be a strictly positiveselfadjoint operator, whose distinct eigenvalues and eigenfunc-tions are > 0i and i, i , respectively. For , we introducethe Hilbert spaces

= ( ) = = H A u v A u A v u A u, , , , ./4 /4 /4 /4

The symbol , will also be used to denote the duality productbetween H and its dual space H . In particular, we have thecompact embeddings +H H1 , along with the generalized Poin-car inequalities

( ) + +w w w H, , 41 4 14 1

and we define the family of product Hilbert spaces

= [ ] + +H H H H , 0, 2 .2 2

In all these notations the index is omitted when = 0. Then, westate on the following abstract Cauchy problem:

( )

+ + + ( + ) + ( ) =

+ + + ( + ) ( ) = 5

u Au u u A u u u f

u Au u u A u u u f

,

,

tt t

tt t

1 1 1 1 12 1/2

1 1 2 1

2 2 2 2 12 1/2

2 1 2 2

( ( ) ( ) ( ) ( )) = ( ) ( )u u u u u v u v0 , 0 , 0 , 0 , , , . 6t t1 1 2 2 10 10 20 20

The original problem (1)(3) can be viewed as a special case of(5)(6) by assuming = A xxxx and = ( )H L L0,2 . We stress that thisabstract formulation cannot be applied when boundary conditionsdiffer from (2) (for instance, if clampedclamped or hingedclamped ends are prescribed). Really, the original coupled systemcan be described by means of a single operator A only if the beamsare assumed to be hinged at their ends. Afterwards, a weak solu-tion of (5)(6) will be denoted by +: ,

( ) = ( ( ) ( ) ( ) ( ))t u t u t u t u t, , , .t t1 1 2 2

For further convenience, (5) may be rewritten as a system witha symmetric nonlinear coupling term independent of . Indeed,letting

= ( + ) = ( ) = ( + ) = ( )w u u v u u f f f g f f, , , ,12 1 2

12 1 2

12 1 2

12 1 2

(

I. Bochicchio et al. / International Journal of Non-Linear Mechanics 85 (2016) 161173 163

(5) takes the following form:

( )

+ + + + ( + + )

+ ( + ) =

+ + + + + ( + + )

+ ( + ) =

7

w Aw w w v w v A w

w v w v A v f

v A I v v w v w v A v

w v w v A w g

2

2,

22

2.

tt t

tt t

12

12 1/2

12

12 1/2

12

12 1/2

12

12 1/2

2.1. Energy norm and semigroup of solutions

Henceforth, c will denote a generic positive constant whichpossibly (but implicitly) depends on the structural constants of theproblem. In addition, + + Q: 0 will denote a generic increasingmonotone function which explicitly depends only on R, but im-plicitly also depends on the structural constants of the problem.The actual expressions of c and Q may change, even within thesame line of a given equation.

The total energy of a solution s is defined as

( ) ( )

( ( )) = ( ( )) + + ( ) + + ( )

+ ( ) ( )

E t t u t u t

u t u t

12

,

1 12 2

2 12 2

1 22

where represents the energy norm of s in , namely

( ( )) = ( ) = ( ( ) + ( ) ) + ( ) + ( )t t u t u t u t u t .t t2 1 22 2 22 1 2 2 2

After assuming that a continuous solution ( )t , t 0, does exist,suitable identities and energy estimates are obtained. They all arewritten here in a formal setting and could be made rigorous withthe use of mollifiers (a procedure which involves boring androutine calculations). The very same identities and estimates applyto the Galerkin approximation scheme which is required to proveexistence.

The energy identity is (formally) obtained after multiplying (5)1by ut 1 and (5) 2 by ut 2. It reads

( ( )) + ( ( ) + ( ) )

= ( ( ) + ( ) ) ( )

ddt

E t u t u t

u t f u t f

2

2 , , . 8

t t

t t

12

22

1 1 2 2

Lemma 1. Let fi H 2, i1,2, and , >, , , 0. For all >t 0and initial data = ( ) z u v u v, , ,10 10 20 20 with z R, we have

( ( )) ( )t Q R .

Proof. We introduce the functional : given by

( ) ( ) = ( ) + ( )E u f u f2 , , . 91 1 2 2Along any solution ( )t to (5), the time function ( ( ))t is non-increasing. Actually, from the energy identity (8) it follows

( ( )) = ( ( ) + ( ) ) ( )ddt

t u t u t2 0, 10t t12

22

which ensures that

( ( )) ( ) ( )t z Q R ,

for all z with z R. Since E , from (9) we obtain theestimate

( ) + ( + ) u f u f f f2 , , 2 ,1 1 2 2 12 1 22 2 22which finally yields

( ( )) ( ( )) + ( + ) ( ) t t f f Q R2 4 .1 22

2 22

Lemma 2. Let f Hi 2, i1,2, and , >, , , 0. For all initial

data z , the abstract Cauchy problem (5)(6) admits a uniquesolution ( )T0, ; , which continuously depends on the initialdata.

The proof is omitted in that it can be obtained by parallelingProposition 1 in [9], and relies on a standard FaedoGalerkin ap-proximation scheme (see [2,3]) together with a slight general-ization of the usual Gronwall lemma. In particular, the uniform-in-time estimates needed to prove the global existence are exactlythe same as in Lemma 1.

According to Lemma 2, system (5)(6) generates a stronglycontinuous semigroup (or dynamical system) S(t) on , i.e. for agiven initial data z ,

( ) = ( ) ( ) = ( ) t S t z t S t z, .2

Moreover, S(t) admits a Lyapunov functional, according to thefollowing result.

Lemma 3. If f Hi 2, i1,2, then , defined in (9) is a Lyapunovfunctional for S(t), that is a function ( )C , satisfying the fol-lowing conditions:

(i) ( ) z if and only if z ;(ii) ( ( ) )S t z is nonincreasing for any z ;iii) ( ( ) ) = ( )S t z z for all >t 0 implies that z , where

= { ( ) = > }z S t z z t: , 0

is the set of all stationary solutions.

Proof. By the continuity of , and by means of the estimates

( ) ( ) ( ) + E z c z E z c z, ,12

32

assertion (i) can be easily proved. The nonincreasing monotonicityof , along the trajectories departing from z, namely (ii), has beenyet shown in the proof of Lemma 1. Finally, if is constant in time,from (10) we have ( ) = ( ) =u t u t 0t t1 2 for all >t 0, which impliesthat ( ) = u t u1 1, ( ) = u t u2 2 are constants and that they satisfy (1).Hence, = ( ) = ( )z S t z u u, 0, , 01 2 for all >t 0, and then z .

3. Steady states

In this section we analyze the stationary problem related to (5)in the homogeneous case ( = )f f 01 2 . Depending on the modelparameters and , there exist nontrivial (buckled) solutions, bothunimodal and bimodal. In spite of the perfect symmetry of thesystem, for special values of the parameters we obtain non-sym-metric static solutions where the elastic energy is not evenlydistributed between the two beams. The proof of these results canbe found in [16] and is summarized in the Appendix. The non-homogeneous case can be handled according to [15].

Let 0 be the set of stationary solutions to the homogeneousproblem. It is made of vectors = ( ) z u u, 0, , 00 1 2 such that( ) u u H H,1 2 2 2 are weak solutions to the system

+ ( ) + ( ) =+ ( ) ( ) = ( )

Au C u A u k u u

Au C u A u k u u

0,

0, 11

1 1 1 11/2

1 1 2

2 2 2 11/2

2 1 2

where

( ) = ( + ) = =C u u k i/ , / , 1, 2.i i i1 12

In particular, it reduces to

+ ( ) = ( ) =( ) = ( ) = ( ) = ( ) = = ( )

u C u k u u

u C u k u u

u u L u u L i

0,

0,

0 0 0, 1, 2, 12i i i i

1 1 1 1 2

2 2 2 1 2


when = A xxxx and = n L/n 4 4 4, ( ) = ( )x n x Lsin /n . Of course, thetrivial solution = =u u 01 2 does always exist.

3.1. Unimodal solutions

First, we look for solutions where each field involves only oneeigenfunction, namely we consider

( ) = ( ) ( ) = ( ) u x a x u x b x m n, , , .n n m m1 2

After replacing these expressions, from (12) we achieve the system

( + + ) =

( + + ) = ( )

C k a kb

C k b ka

0,

0, 13

n n n n m m

m m m m n n

1

2

where

= + = +

C

La C

Lb

2,

2.n n m m1

22

2

When m n, it easily follows that = =a b 0n m and = =u u 01 2 .Accordingly, if a nontrivial solution ( )u u,1 2 of this kind does existthen it is unimodal, in the sense that both u1 and u2 should involvethe same eigenfunction,

( ) = ( ) ( ) = ( ) ( )u x a x u x b x n, , . 14n n n n1 2

In the sequel, we consider a fixed n .

Theorem 1. Depending on the values of , there exist at mostnine unimodal solutions to (11) involving the n-th eigenfunction of A.In particular:

when n there exists only the null solution( ) = ( )( ) ( )a b, 0, 0n n0 0 ; at = n the null solution bifurcates into a couple of in-phase solutions,

( ) = ( )( ) ( )a b, , ;n n n n1 1 at = k2 /n n the null solution bifurcates into a couple

of out-of-phase solutions,

( ) ( ) = ( ) ( )a b, , ,n n n n2 2 at = k3 /n n each out-of-phase solution bifurcates

into a couple of solutions with unequally distributed energy,namely

( ) ( ) ( ) ( ) = = ( ) ( ) + ( ) ( ) +a b a b, , , , , .n n n n n n n n3 3 4 4

Fig. 3. The energy level sets of E at

Summarizing, there are

one solution (the null solution) in the range / n ; three solutions in the range ( ) < k2 / /n n n ; five solutions in the range ( ) < ( )k k3 / / 2 /n n n n ; nine solutions in the range < ( )k/ 3 /n n .

Remark 1. This situation occurs also in the study of buckling for arectangular plate under in-plane loading. This phenomenon isnamed mode jumping: its buckling pattern exhibits a symmetricbreaking (secondary) bifurcation, that is the symmetry group ofthe equations remains unchanged, but the solutions which bi-furcate break the symmetry (see [34, pp. 149, 242]).

By replacing (14) into the expression of the total energy E, we get

( ) = ( + ) + + + +

+ ( )

E a b a b a b

a b

,12

12n n n n n n n n n

n n

2 2 2 22

22

2

and using the level-set representation we infer that (seeFigs. 3 and 4)

in the range n the null solution is the unique mini-mum of E;

in the range < k2 /n n n the energy E admitstwo minima and a saddle point (the null solution);

in the range < k k3 / 2 /n n n n the energyE admits two minima, two saddle points and a maximum (thenull solution);

in the range < k3 /n n the energy E admits fourminima, four saddle points and a maximum (the null solution).

If we restrict our attention to the amplitude an of u1, Fig. 5shows the corresponding bifurcation diagram depending on thevalues of = qn n .

Remark 2. After the secondary bifurcation, the double-beamsystem exhibits steady-states where the beams have differentelastic energies. Such non-symmetric equilibria correspond tounstable branches and cannot be observed. In spite of this, theiroccurrence is relevant in the longtime global dynamic of the twobeams, for it may lead up to non-symmetric energy exchanges

= 2.5, 3.5 ( = = = = 1n ).

Fig. 4. The energy level sets of E at = 5 ( = = = = 1n ).


between them.

3.2. Bimodal solutions

It is well known that all the steady states of a single extensiblebeam are expressed as unimodal functions (see, for instance[7,15]). We scrutinize here whether this is the case even for thedouble-beam system under consideration. At a first attempt, weare looking for bimodal solutions. Of course, if solutions of thiskind do exist, they cannot take the in-phase shape because of thesystem uncoupling. A more general approach is devised in [16].

3.2.1. The resonant setLet

= ( ) = + n, 2 / , ,n n n n n

which represent the eigenvalues of the operators A1/2 and + A A21/2 1/2, respectively. According to the previous section,system (12) exhibits unimodal in-phase bifurcation when = nand unimodal out-of-phase bifurcation when = ( )n .

Without any loss of generality, we assume m n, , >m n.Then >m n, but ( )n could overlap either m or ( )m if isproperly chosen. In the former case,

( ) = =

2 .n mn m mn n m n

In the latter case,

Fig. 5. The buckled solutions ( )ani , =i 1, 2, 3, 4, for a fixed n .

( ) = ( ) = 2n mn m mn mn n m

and then ( ) = +n mn n m. The set of all the values of thatproduce some overlapping is named resonant set:

{ } = = = >+ m n: either or , for some integer .mn mnNow, we look for nontrivial bimodal solutions of the form

( ) = ( ) + ( ) ( ) = ( ) + ( )u x a x c x u x b x d x, ,n n m m n n m m1 2

where a b c d, , , 0.n n m m When both beams have the same elasticenergy, the following result holds (see the Appendix).

Theorem 2. Given >m n, bimodal solutions such that = u u1 1 2 1 are allowed only if . In particular,

when = mn there exist 1 bimodal solutions, where both modesare out-of-phase, provided that < ( + )n m ; when = mn there exist 1 bimodal solutions, where the m-modeis in-phase and the n-mode is out-of-phase, provided that

< m.

It is apparent that after bifurcation infinitely many (n,m) bimodalsolutions just occur instead of n and m unimodal ones when theircritical values n and m overlap.

A different result holds if the total elastic energy at equilibriumare unequally distributed among the beams. After introducing = >/ 1mn m n and

( ) = ( + ) + ( )( )( ) ( )

1 1 1 2 /

1 2 /,

15mnmn mn mn

mn mn

2

( ) = ( + ) ( )( ) ( )

1 1 1 2 /

1 2 /,

16mnmn mn mn

mn

2

we are able to prove (see the Appendix).

Theorem 3. Given >m n, four distinct bimodal solutions such that u u1 1 2 1 exist if and only if either

< < ( ) < < ( ) and2 ,mn mn 1 2

or

< < ( ) and, ,mn 1

where

( ) = ( ) ++ ( )

( ) = ( ) ++ ( ) 1 ,

1.n mn

nm

mn

m1 2

4. The exponential decay

Within this section, we restrict our attention to the homo-geneous system, namely we assume fi0 into (5). It is easy tocheck that the linear part of the differential operator acting on uifulfills the following lemma.

Lemma 4. (See [6, Lemma 4.5]) Let ( ) = + u Au A ui i i1/2 ( = )i 1, 2and

=

+ < 1 .

As a consequence, letting ( )S t0 be the semigroup which is generatedby (5) when fi0, we are able to prove the following result.

Theorem 4. Let z , z R. Provided that > 1 , allsolutions ( )S t z0 decay exponentially, i.e. there exists a positive con-stant c such that

( ) ( ) t Q R e .ct

Proof. We introduce the functional

= + + ( )

E u u u u, ,1

,18

t t1 1 2 22

and we choose

=( )

C

Cmin 1, , ,

219

11

Taking into account (17), this constant is positive provided that > 1 . After recalling the definition of E, we remark that

( )

= ( ) + ( ) + +

+ + + +

+ ( )

u u u u u u

u u u u u u

u u

, ,

, ,

. 20

t t

t t

1 1 2 2 12

22

12 1 1

42 1

41 1 2 2

1 22

The first step is to prove the equivalence between and, thatis

( ) ( )C Q R

2. 21

Indeed, a lower bound is provided by virtue of (19) and Lemma 4,

+ + +

u u C u u

C

12 2

2.

t t12

22

11 2

22 2

2

On the other hand, by applying Young inequality and Lemma 1tailored to ( )S t0 , the upper bound of follows:

+ + || + + ( ) = ( )

Q RQ R2

2 2.

1 1

2

1 12

The last step is to prove the exponential decay of. To this aim,we obtain the identity

+ + ( )( + ) + ( + )

+ ( )( + ) =

ddt

u u u u

u u u u

22

, , 0,

t t

t t

12

22

1 14

2 14

1 1 2 2

where is given by (19). Exploiting (4), (21) and the Young in-equality, we have

+ + ( )( + ) ( )

+

ddt

u u u u

C

4

2,

t t12

22 2

11 2

22 2

2

2

1

from which it follows

+

ddt C

C2

2 0.1

1

Letting = c C C2 /21 1 , which is positive by virtue of(19), we conclude

( ) ( ) ( ) ( ) C t t e Q R e2

0 .ct ct

5. The global attractor

The existence of a Lyapunov functional (see Lemma 3) ensuresthat bounded sets of initial data have bounded orbits. In thissection, we provide the existence of the global attractor byshowing a suitable (really, exponential) asymptotic compactnessproperty of the semigroup. Namely, we prove the following.

Theorem 5. Let =f H i, 1, 2i 2 , be fixed. Then, the semigroup S(t)acting on possesses a connected global attractor . In par-ticular, is bounded in 2, so that its regularity is optimal.

By standard arguments of the theory of dynamical systems (see[1,20,33]), the first part of Theorem 5 can be established by ex-ploiting the following:

Lemma 5. Assume that S(t) fulfills the asymptotic compactnessproperty. That is, for every >R 0, there exist a function + + :Rvanishing at infinity and a compact set R such that the semi-group S(t) can be split into the sum ( ) + ( )L t K t , where the one-parameter operators L(t) and K(t) fulfill

( ) ( ) ( ) L t z t K t z, ,R R

whenever z R and t 0. Then, S(t) possesses a connectedglobal attractor R0 for some >R 00 .

According to the scheme first devised in [19], we split the so-lution ( )S t z into the sum

( ) = ( ) + ( )S t z L t z K t z,

where

( ) = ( ( ) ( ) ( ) ( ))

( ) = ( ( ) ( ) ( ) ( ))

L t z u t u t u t u t

K t z u t u t u t u t

, , , ,

, , ,

Lt

L Lt

L

Kt

K Kt

K

1 1 2 2

1 1 2 2

respectively solve the systems

( )( )

+ + + + + =

+ + + + + =

( ( ) ( ) ( ) ( )) = ( )

u Au u u A u au

u Au u u A u au

u u u u z

0

0

0 , 0 , 0 , 0 , 22

ttL L

tL L L

ttL L

tL L L

Lt

L Lt

L

1 1 1 1 12 1/2

1 1

2 2 2 2 12 1/2

2 2

1 1 2 2

( )

+ + + ( + ) + ( ) =

+ + + ( + ) ( ) =

( ( ) ( ) ( ) ( )) =

23

u Au u u A u au u u f

u Au u u A u au u u f

u u u u

,

,

0 , 0 , 0 , 0 0,

tt K K t K K L

tt K K t K K L

K t K K t K

1 1 1 1 12 1/2

1 1 1 2 1

2 2 2 2 12 1/2

2 2 1 2 2

1 1 2 2

where >a 0 is large enough that the following inequality holds(see [19])

+ + ( )u u u a u m u 2412 2

222

12 2

22

for all u H2 and for some = ( ) m m a, 1.The proof of Lemma 5 is carried out in the next two subsec-

tions, and the last part of Theorem 5 follows from the compactembedding 2 .

5.1. The exponential decay of L(t)

We prove here the following lemma.

Lemma 6. Let z R. There is = ( ) >c c R 0 such that

( ) ( ) L t z Q R e .c t


Proof. Let 0 be the energy norm of the one-parameter operator L(t), namely

( ( ) ) ( ) = ( ( ) + ( ) ) + ( )

+ ( ) ( )

L t z L t z u t u t u t

u t . 25

L Lt

L

tL

02

1 22

2 22

12

22

We introduce the functional

( )( )( )

( ) = ( ( ) ( ) ( )) = ( ( ) )

+ ( ( ) + ( ) ) + ( ) + ( )

+ ( ) ( ) + ( ) ( )

+ ( ) ( ) + ( ) ( )

t L t z u t u t L t z

u t u t a u t u t

u t u t u t u t

u t u t u t u t

, ,

, , .

L L L L

L L

tL L

tL L

0 0 1 2 0

1 12

2 12

12

22

1 12

1 12

2 12

2 12

1 1 2 2

Using Lemma 1 and (24), we obtain the following lower andupper bounds for 0:

( ) ( )Q R . 2614 0 0 0

The time-derivative of 0 along a solution to system (22) reads

+ + ( ) +

= +

( ) +

ddt

u u

u A u u u A u u

u u u u

2

2 , ,

, , ,

tL

tL

tL

tL

Lt

L Lt

L

0 0 12

22

11/2

1 1 12

21/2

2 2 12

1 1 2 2

where the second term can be estimated in terms of 0 as follows:

( )

( )

+ + ( )

+ ( )

( ) + ( ) + ( ) +

u A u u u u

u u A u u u

u

Q R u Q R u

2 , ,

1.

t i i iL

iL

t iL

t i iL

i iL

t iL

iL

t i t iL

1/212

21

222

2

1

1/2 222 2

2 2

2 02

02

2

10

Choosing as small as needed in order that ( ) + 0 and t s 0,

( ) ( ) + ( ) + ( )u y u y dy t s Q R .s

t

t t1 2

The exponential decay of 0 is then entailed by exploiting ageneralized Gronwall Lemma (see [12, Lemma 2.1, 13, Lemma 3.7]).Finally, from (26) the desired decay of 0 follows.

5.2. The compactness of R

Now, we prove the existence of a compact set R which con-tains all orbits ( )K t z , z R, t 0. This result is achieved bytwo steps: first the boundedness of these orbits in a more regularspace, 2, is proved in Lemma 8, then the compactness of R isensured by the compact embedding 2 (see [9,10]).

Lemma 8. Let =f H i, 1, 2i 2 , be fixed and let z , z R.Then, there exists >K 0R such that

( ) ( )K t z K t, 0. 27R2

Proof. Let introduce the energy norm of order 2 of K(t), namely

( )= + + + u u u u .K K K t K t K1 42 2 42 1 22 2 42For any ( )0, 1 we define the functional

( )= + + + J F u u u u2 , , ,K K K t K K t K K1 1 2 2 2 2where

( ) ( )( )

= + ( ) + + ( )

+

F u t u u t u

f u f u2 , , .

K K K

K K

1 12

1 32

2 12

2 32

1 1 2 2 2 2

Choosing small enough, the following bounds hold true:

( ) + ( ) ( )Q R J Q R2 . 28K K K12

We estimate the time derivative of JK along a solution ( )K t z . To thisend we first compute

+ = +

+ +

+ +

( )

ddt

F u u

a u u u u

u u u u u u

u u u u

2

2 , ,

2 , ,

2 , .

K Kt

Kt

K

tK L

tK L

Kt

Kt

tK K

1 22

2 22

1 1 2 2 2 2

1 32

1 1 1 2 32

2 2 1

1 2 1 2 2

We deal with each term at the r.h.s. separately. By virtue of theinterpolation inequality

( ) + , , 291 1

taking into account (4) and Lemma 1, we easily obtain ( = )i 1, 2 ,

( )

+ ( ) +

u u u u u u Q R u

a u uc

uQ R

,1

,

,16 16

.

iK

i t i iK

i t i t iK

t iK

iL

iL K K

32

11

42

2

2 22

Analogously

( ) ( ) +u u u u Q R,8

.t K K K1 2 1 2 2

In addition, a straightforward calculation leads to

+

= + +

+ + +

+ +

+ +

ddt

u u u u

u u u u

u u u u a u u u u

u f u f u u u u

u u u u

, ,

, , , ,

, , ,

.

tK K

tK K

tK

tK K K

tK K

tK K K L K L

K K K K

K K

1 1 2 2 2 2

1 22

2 22

1 42

2 42

1 1 2 2 2 2 1 1 2 2 2 2

1 1 2 2 2 2 1 2 1 2 2

1 12

1 32

2 12

2 32

From interpolation inequality (29), we have

( )u u u u u u u u Q R, ,K K K K K1 2 1 2 2 1 2 1 1 2 3

hence


+

( + ) + + + ( )

+ ( )

ddt

u u u u

u u u u Q R

Q R

, ,

12

.

tK K

tK K

tK

tK K K

K

1 1 2 2 2 2

1 22

2 22

1 42

2 42

Collecting all these inequalities we end up with

+ ( + ) + + +

( ) + ( ) + + ( )

ddt

J u u u u

Q R Q R u u Q R

2 1

2.

Kt

Kt

K K K

K K Kt t

1 22

2 22

1 42

2 42

1 2

Choosing now = > 00 small enough, in light of (28) thisturns into

+ ( ) + ( ) + + + ( )

ddt

J J Q R J Q R u u J Q R2

1 .K K K t t K0

1 2

Since ( ) =J 0 0K , once more from Lemma 7 and the generalizedGronwall Lemma (see [12, Lemma 2.1, 13, Lemma 3.7]) we get thethesis.

5.3. The structure of the attractor

As well known [1], when a finite set of stationary solutionsoccurs, the global attractor of the semigroup S(t) consists of theunstable manifolds connecting them. Due to Theorems 13, wecan conclude:

Corollary 1. Provided that 1 and , the global at-tractor consists of the unstable manifolds connecting thesteady states of S(t).

Even if we restrict our attention to unimodal solutions (in this casereduces to 4), a complete picture of is hard to achieve.

Really, some informations may be sketched by means of Poincarsections, which are obtained by intersecting trajectories in theoriginal phase-space with a lower dimensional subspace trans-versal to the flow of the system. This technique may be exploitedfor both the damped and undamped ( > 0) system, although it isa routinary tool of the latter case. Unfortunately, there is no gen-eral method to construct a Poincar map.

Here, we may take advantage of the representation (7) of thesystem. Indeed, when = =f g 0 and assuming suitable initialconditions, this system admits special dynamical solutions of theform ( ( ) )w t , 0 and ( ( ))v t0, , corresponding to perfectly symmetric( = u u1 2) and antisymmetric ( = u u1 2) oscillations. If this is thecase, then w and v satisfy the uncoupled first order systems

Fig. 6. A branch of the global attractor (on the left) and basins of attraction (on the rightphase-subspace ( )v ,n n , when = =n 1 ( and are the same as in Fig. 7a).

+ + + [ + ] = = ( )

Aw w A w

w

0

30

t

t

12 1/2

+ + + [ + ] = = ( )

A v v A vv

0

31t

t

12 1/2

and the phase portraits of their unimodal solutions

( ) = ( ) ( ) ( ) = ( ) ( ) w x t w t x v x t v t x n, , , , ,n n n n

can be scrutinized as in [31]. For instance, assuming

< k3 ,n

n

we conclude that in each sub-space ( )w ,n n and ( )v ,n n of the four-dimensional phase-space there is a branch of the attractor ,and the basins of attraction of the stable steady-states relyingtherein may be easily depicted (see Fig. 6).

6. Conclusions

In this paper we scrutinized both statics and dynamics of adamped elastically coupled WK double-beam system under evencompressive axial loading.

A complete characterization (in closed form) of solutions to thenonlinear stationary problem with vanishing sources is achieved.As well as the null solution, the system may exhibit unimodal(only one eigenfunction is involved) and bimodal (two eigen-functions are involved) buckled solutions. In Theorem 1 thenumber of unimodal solutions is proved to depend on the ratios

/ and / . In spite of the perfect symmetry of the system, forspecial values of the parameters we obtain non-symmetric staticsolutions where the elastic energy is not evenly distributed be-tween the two beams (see also the Appendix, Fig. 8).

Since the axial displacement D and the thickness h are com-parable and considerably shorter than the length L and the spacingd of the beams, under reasonably physical assumptions on thestiffness of the elastic core we may conclude that || and sharethe same order of magnitude h L/ , whereas is much smaller.Accordingly, the order of magnitude of | |/ and / is L h/ 1.Hence, inequalities on n in Theorem 1 are not narrowed by themodeling assumptions and all the stationary solutions exhibitedhere are physically consistent.

The complete analysis of buckling bifurcations (see Fig. 5) andthe related picture of the energy level sets (see Figs. 3 and 4) re-veal that stability of unimodal stationary solutions strongly de-pend on the coupling constant . This is not surprising whennonlinear equations are involved. In nonlinear electroelastostatics,

) in the phase-subspace ( )w ,n n , when = =n 1. The same picture holds true in the


for instance, the stability of steady states may be crucially de-pendent on the magnitudes of the electromechanical couplingparameters (see [17]).

The occurrence of a so complex structure of the steady statesmotivates a global analysis of the longtime damped dynamics.When / is larger than the first eigenvalue 1 of the Laplacianoperator A1/2, then in Theorem 4 we prove exponential stability ofthe (unique) trivial solution. On the contrary, when 1buckled steady-states occur and we prove in Theorem 5 the ex-istence of a global regular attractor of dynamical solutions. If inaddition , the resonant set defined in Section 3.2.1, then thereexists a finite number of stationary solutions and the global at-tractor consists of the unstable manifolds of trajectoriesconnecting them (see Section 5.3 and figures therein).

Appendix A

A.1. Proof of Theorem 1

When mn from (13) we get the system

= ( + )

= ( + ) ( )

b a p a

a b p b

,

, 32

n n n n n

n n n n n

2

2

where

= + + =

p L

L2,

2.n

nn

nn

n

Assuming a b, 0n n and putting = +Y p an n n2, from (32) it follows

+ =Y p Y

pY

10.n n n

n

nn

n

4 32 4

Accordingly, this equation can be written as a product,

+ =

Y Y p Y

1 10,n

nn n n

n

22

22

so yielding four solutions

= = ( ) ( )Y Y p p1/ , 4 /2 . 33n n n n n n n n2 2

Letting = qn n and

=

=

= +

Lq

Lq

Lq q q

2,

2 2,

1 3,

nn

n

nn

n n

nn

n n

n

n n

n

n n

from (32) and (33) we obtain the following expressions for theamplitudes an and bn:

( ) = ( ) >

( ) = ( ) >

( ) = ( ) >

( ) = ( ) >( )

( ) ( )

( ) ( )

( ) ( ) +

( ) ( ) +

a b q

a b q

a b q

a b q

, , if 0,

, , if2

,

, , if3

,

, , if3

,34

n n n n n

n n n n nn

n n n n nn

n n n n nn

1 1

2 2

3 3

4 4

which provide all real unimodal solutions involving the n-theigenfunction.

When n1 and < 3 /1 1 , these solutions are

represented in Figs. 7 and 8 (the subscript 1 is omitted). Thebuckling modes here depicted resemble a study carried out in [28]within the framework of the buckling localization.


From (12) we achieve

[( + + ) ] + [( + + ) ] =

[( + + ) [ + [( + + ) ] =

C k a kb C k c kd

C k b ka C k d kc

0,

0,

n n n n n m m m m m

n n n n n m m m m m

1 1

2 2

where

= + = + =

= + = + =

CL

X X a cL

u

CL

X X b dL

u

2,

2,

2,

2.

n n m m

n n m m

1 1 12 2

1 12

2 2 22 2

2 12

Because of the orthogonality of distinct eigenfunctions, we obtainthe system

= [ + + + ]

= [ + + + ]

= [ + + + ]

= [ + + + ]

kb a k L X

kd c k L X

ka b k L X

kc d k L X

/ /2 ,

/ /2 ,

/ /2 ,

/ /2 .

n n n n n

m m m m m

n n n n n

m m m m m

1

1

2

2

For the sake of simplicity, we let

= + + = =

L

Lj m n

2,

2, ,j j

jj

j

and we rewrite the previous system in the compact form

= ( + )= ( + )= ( + )= ( + ) ( )

b a X

d c X

a b X

c d X

,,

,, 35

n n n n

m m m m

n n n n

m m m m

1

1

2

2

from which it follows

= ( + )( + )= ( + )= ( + )( + )= ( + ) ( )

a a X X

b a X

c c X X

d c X

,

,

,

. 36

n n n n n

n n n n

m m m m m

m m m m

21 2

1

21 2

1

From the first and third equations of (36) we obtain a system in-volving X1 and X2, only,

= ( + )( + )

= ( + )( + ) ( )

X X

X X

1 ,

1 . 37

n n n

m m m

21 2

21 2

If it admits positive solutions ( ) X X,1 2 , then the four amplitudes ofthe bimodal solutions u1 and u2 follow by solving

= ( + )

= ( + )

+ =

+ = ( )

b a X

d c X

a c X

b d X

,

,

,

. 38

n n n n

m m m m

n n m m

n n m m

1

12 2

12 2

2

When the beams have the same elastic energy, system (37) pro-vides

=

= ( )

X

X

,

. 39

n n

m m

1

1

These relations are satisfied provided that

Fig. 7. Symmetrical in-phase (a) and out-of-phase (b) static solutions (n1).

Fig. 8. Asymmetrical out-of-phase static solutions (n1).


= ,n n m m1 1

which leads to the following four occurrences:(a) = n n m m1 1 , so that =n m which contradicts the as-

sumption >m n and then no bimodal solution exists;(b) + = + n n m m1 1 , from which it follows = mn and then

( ) = ( )n m . If this is the case, (38) takes the form

( )( )

= =

+ = + + )

+ = + + )

b a

d c

a c L

b d L

,,

2 / ,

2 / ,

n n

m m

n mn m m n n

n mn m m n n

2 2

2 2

where = >/ 1mn m n , and admits real solutions providedthat

< ( + ) = ( ) .m n n mn

Letting = = = =a b U c d V,n n m m and

() = + +

L

2,

nm n

2

we obtain the following (elliptic) relation between the ampli-tudes

+ = ()U V .mn2 2 2

Accordingly, in this case there exist infinitely many out-of-phase solutions;

(c) = n n m m1 1 , from which it follows = ( )/2n m m : since must be positive, its expressioncontradicts the assumption >m n and then no bimodal solu-tion exists;

(d) + = + n n m m1 1 , from which it follows = mn and then ( ) =n m. If this is the case, (38) takes the form

( )( )

= =

+ = +

+ = +

b a

d c

a c L

b d L

,,

2 / ,

2 /

n n

m m

n mn m m n

n mn m m n

2 2

2 2

and admits real solutions provided that

< = ( ) .m n mn


Letting = = = =a b U c d V,n n m m and

() = +

L

2,

nm

2

we obtain the following (elliptic) relation between the ampli-tudes

+ = ()U V .mn2 2 2

Accordingly, in this case there exist infinitely many asym-metric solutions.


To prove this theorem, first we establish a preliminary result.Letting

= ( + ) = ( + ) ( )x X y X, , 40n n n n1 2

= ( + ) = ( + )w X z X, ,m m m m1 2

we obtain

= = ( ) = ( )( )x w y z 1 1 2 / .mn mn m n m mn mn

From these relations and (37) we obtain the system

==

= ( )( ) = ( )( ) ( )

xy

wz

x w

y z

1,1,

1 1 2 / ,

1 1 2 / . 41

mn mn mn

mn mn mn

We stress that the coefficients involved in this representation areindependent of L, and, especially, on .

Lemma 9. Let >m n be given, if and only if either > 2 mn or < < 0 mn or < 2 mn, then ( ) > ( ) > 2mn mn . Solutions to (42) arepositive pairs. In addition, ( ) > ( ) > > ( ) > ( ) > w x y z1 0(see Fig. 9).

When X 02 ,

can be established from (40) in dependence of the values of and,

= + ( ) + = + ( ) +

>

= + ( ) + = + ( ) +

>

XL

xL

w

XL

yL

z

2 1 2 1

0,

2 1 2 1

0.

nn

mm

nn

mm

1

2

Since ( ) ( ) y x and ( ) ( ) z w for all , these conditions areboth satisfied provided that < ( ) , where

( ) = ( ) = ( ) y z1 1

nn

mm

or alternately

( ) = ( ) + + ( ) = ( ) + + ( ) y z1 1

.nn

mm

We note that ( ) = ( ) = n m n and ( ) = ( ) = n m.

According to the previous analysis, from (38) the system in-volving the unknown amplitudes takes the form

Fig. 9. The graphic solution of subsystems (42) when > 2 .

Fig. 10. The graphic solution of subsystems (42) when 2 mn, then < ( ) < < ( ) < ( ) y x w0 1 and

+

w xr1 1

1 1 .2 2

2

2

As a consequence,

+ w x

r2

2

2

22 2

and then no solution exists. When


< + ( )

xx y

w0 .

n2 2

From the left inequality we obtain 1, where ( = y x1/ )

= ( )

= + ( )

x x y

xx

1,

n n1

2

2

and then

( ) = ( ) ++ ( ) 1 .n mn

n1

From the right inequality and using (41)3 we have 2, where( = z w1/ )

= + ( )

= + ( ) <

w z w

ww

1m m2

2

2

and then

( ) = ( ) ++ ( ) 1 .m mn

m2

Summarizing, solutions exist if and only if 1 2. When < 0 mn, then ( ) < < ( ) < < < ( ) y x w1 0 1 ,which implies

+w

r .2

22 2 2

As a consequence, the condition to have intersections reduces to

+

rx

2 22

2

and the following condition for follows:

+ ( )

x y

x,

n2

from which we obtain 1.

We finally observe that

( ) = +

( ) 1n

mn

n1

and

( ) = =

( ) = =

( ) = = ( + )

lim32

,

lim32

,

lim12

.

n mmn

nn m

m nmn

mm n

nmn

nn m

1

2

1

mn

mn

mn

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Buckling and nonlinear dynamics of elastically coupled double-beam systemsIntroductionPreliminary resultsEnergy norm and semigroup of solutions

Steady statesUnimodal solutionsBimodal solutionsThe resonant set

The exponential decayThe global attractorThe exponential decay of L(t)The compactness of KRThe structure of the attractor

ConclusionsProof of Theorem 1Proof of Theorem 2Proof of Theorem 3

References

international journal of non-linear mechanics · 162 i. bochicchio et al. / international journal...

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