spectralanalysis, stability andbifurcations systems nonlinear...

Nonlinear PhysicalSystems

Spectral Analysis, Stability and Bifurcations

Edited byOleg N. Kirillov

Dmitry E. Pelinovsky

Series EditorNoel Challamel

iSfE WILEY

First published 2014 in Great Britain and the United States by ISTE Ltd and John Wiley & Sons, Inc.

Apart from any fair dealing for the purposes of research or private study, or criticism or review, aspermitted under the Copyright, Designs and Patents Act 1988, this publication may only be reproduced,stored or transmitted, in any form or by any means, with the prior permission in writing of the publishers ,or in the case of reprographic reproduction in accordance with the terms and licenses issued by theCLA. Enquiries concerning reproduction outside these terms should be sent to the publishers at theundermentioned address:

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© ISTE Ltd 2014The rights of Oleg N. Kirillov and Dimtry E. Pelinovsky to be identified as the author of this work havebeen asserted by them in accordance with the Copyright, Designs and Patents Act 1988.

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Preface .

Chapter 1. SurprisingDavide BIGONI, Diego

1.1. Introduction .

1.2. Buckling in ten

1.3. The effect of cc

1.4. .The Ziegler per

1.5. Conclusions .

1.6. Acknowledgme1.7. Bibliography.

MIX

British Library Cataloguing-in-Publication DataA CIP record for this book is available from the British LibraryISBN: 978-1-84821-420-0

A@" -uJ Paper fromFSC responsible sources

wwwfscorg FSee C013604

Printed and bound in Great Britain by CPI Group (UK) Ltd., Croydon, Surrey CRO 4YY

Chapter 2. WKB Soluand Applications . .Jean-Francois BONY, ~

2.1. Introduction .

2.2. Connection of I

2.2.1. A model in

2.2.2. Classical m

2.2.3. Review of ~

2.2.4. The microh2.2.5. The microl.

2.3. Applications to

2.3.1. Spectral pn2.3.2. Resonance

2.4. Acknowledgrm2.5. Bibliography.

Chapter 1

Surprising Instabilities of SimpleElastic Structures

In this chapter, examples of structures buckling in tension are presented, where nocompressed elements are present, slightly different from those previously proposedby the authors. These simple structures exhibit interesting postcritical behaviors; forinstance, multiple configurations of vanishing external force are evidenced in onecase. Flutter instability as induced by dry friction is also considered in the Zieglerpendulum, with the same arrangement presented by Bigoni and Noselli [BIG 11], butnow considering the dynamical effects due to the mass of the wheel, which waspreviously neglected. It is shown that, for the values of rotational inertia pertinent toour experimental setup, this effect does not change the overall behavior, so thatprevious results remain fully confirmed.

1.1. Introduction

The first example of an elastic structure buckling for a tensile dead load, withoutelements subject to compression, has been provided by Zaccaria et al. [ZAC 11].This finding opens new possibilities in the design of compliant structures. In thischapter, we present a single-degree-of-freedom structure (different from - andslightly generalizing - that found by [ZAC 11]), an example that shows that thepreviously investigated systems are elements of a broad set of structures behaving ina, perhaps, "unexpected way". Moreover, we present a simple generalization of asingle-degree-of-freedom system, further revealing the effects of the constraint's

Chapter written by Davide BIGONI, Diego MISSERONI, Giovanni NOSELLI andDaniele ZACCARIA.

2 Nonlinear Physical Systems

curvature analyzed by Bigoni et al. [BIG 12b]. The presence of an additional springhas an important effect on the post-critical behavior, so that two configurations (inaddition to the trivial one) corresponding to a null external force are found.

Finally, we reconsider the frictional instability setup analyzed by Bigoni andNoselli [BIG 11], where a follower tangential load is transmitted by friction at afreely rotating wheel mounted at the end of a Ziegler pendulum [ZIE 77]. Theapplication of a follower tangential load to a structure was a problem previouslyunsolved [ELI 05, KOI96] , but important from both a theoretical (see, forinstance, [KIR 10]) and applicative point of view (for instance , to energyharvesting [DOA 11]). Within the same setting considered by Bigoni andNoselli [BIG 11], we now analyze the effects on dynamics of the inertia of the wheeland we show that, for the values of inertia pertinent to the experimental setting used,these effects are negligible, so that previous results are now fully confirmed.

~.2. Buckling in tension

Structures buckling under tensile dead loading (without elements subject tocompression) were discovered by Zaccaria et al. [ZAC 11], who pointed out thesimple example of the single-degree-of-freedom system as shown in Figure 1.1.

They also developed the concept by replacing the rigid rods with deformableelements. Though the finding by Zaccaria et al. [ZAC 11] might seem an isolatedcase, we state, on the contrary, that a broad class of structures buckling in tension canbe invented. To substantiate this statement, we provide, as an example, the newsingle-degree-of-freedom system as shown in Figure 1.2, where two rigid rods areconnected through a roller constrained to slide orthogonally to the left rod.

Figure 1.1. A single-deg.under tensile dead loac

For this structure, bifurcation load and equilibrium paths can be calculated byconsidering the bifurcation mode illustrated in Figure 1.2 and defined by the rotationangle ¢ . The elongation of the system and the total potential energy are, respectively,

t1 = l (_1 -1) ,cos ¢

1 2 (1 )W(¢)=-k ¢ -Fl ~-1 ,2 cos ~

[1.1]

so that the force at equilibrium satisfies

F = k¢ cos2

¢.l sin e

[1.2]

Figure 1.2. Bifurcation ofa sing !(the rods of length l are rigid aalong the line orthogonal to the ,attached to the hinge on the left, 1bifurcation and softening in tensdenotes an initial imperfection

Surprising Instabilities of Simple Elastic Structures 3

Figure 1.1. A single-degree-of-freedom structural model showing bifurcationunder tensile dead loading, where two rigid rods are connected through a

slider [ZAC 11J

1.0

..><:

0.6 ~

0.2

0.4

0.8

9060

~o=oo , perfect syst em

3015

, ~-- trivial solutionr

;;.;;.;:,

1'-.,~" ....((.'dO'

o~.~:F,:J; r.... I~;~,~_' --==-....P;;,.:Q-- F

~ <Pcr- ---~/I

!

ebk

r--~--F

[1.1]

(without elements subject toAC 11], who pointed out then as shown in Figure 1.1.

resence of an additional springso that two configurations (inmal force are found.

etup analyzed by Bigoni andis transmitted by friction at agler pendulum [ZIE 77]. TheIre was a problem previouslyboth a theoretical (see, forw (for instance, to energyconsidered by Bigoni and

nics of the inertia of the wheelthe experimental setting used,now fully confirmed.

le rigid rods with deformableC 11] might seem an isolateductures buckling in tension canride, as an example, the new1.2, where two rigid rods arenally to the left rod.

m paths can be calculated by1.2 and defined by the rotationential energy are, respectively,

[1.2]

Figure 1.2. Bifurcation ofa single-degree-of-freedom elastic system under tensile dead loading(the rods of length l are rigid and connected through a roller constrained to smoothly slidealong the line orthogonal to the rigid rod on the left). A rotational elastic spring of stiffness k,attached to the hinge on the left, provides the elastic stiffness. The bifurcation diagram showingbifurcation and softening in tension is reported on the right, where the angle cPo = {1° , 10°}denotes an initial imperfection


1.3. The effect of constraint's curvature

so that the force-rotation relation, shown by the dashed line in Figure 1.2 for ¢o = 1°and ¢o = 10°, is obtained.

In this structure , the constraint is assumed to be smooth and described in the X I -X2

reference system as X 2 = l f( 'ljJ), with 'ljJ = x - ] ] E [0,1] and f'(O) = o.

Figure 1.3. Post-criticastructure. The structur

[1.3]

1 2 (1 1)W(¢ ,¢o) = -k(¢-¢o) -Fl ---- ,2 cos ¢ cos ¢o

F = k (¢ - ¢o) cos2 ¢l sin ¢ ,

Bifurcation loads can be calculated by considering a deformed mode defined bythe rotation angle ¢, assumed to be positive when clockwise. The potential energy ofthe system is

Analysis of the second-order derivative of the strain energy reveals that the trivialsolution is stable up to the critical load Fer = kll , while the non-trivial path,evidencing softening, is unstable. For an imperfect system, characterized by an initialinclination of the rods ¢o, we obtain

The strong effects related to the curvature of the profile on which a structure endis constrained to slide have been highlighted by Bigoni et al. [BIG 12b], who showedhow to exploit a constraint to induce two critical loads (one in tension and one incompression) in a single-degree-of-freedom elastic structure. This structure, as shownin Figure 1.3, can be easily generalized by including an additional elastic spring onthe hinge sliding along the profile, as shown in Figure 1.4.

[1.4]

so that the axial force at equilibrium becomesWhen the profile of the c

signed curvature X= i' 1[1 +and 1.6, the axial load at equi

F = _ kl ¢ + k2 (3 (¢ ) (3' (¢ ) .l [sin ¢ + cos ¢ f' (sin ¢ )]

[1.5]kl ¢V1

F=------l sin ¢(X cos ¢

n energy reveals that the trivial'l , while the non-trivial path,tern, characterized by an initial

[1.3]

I line in Figure 1.2 for ¢o = 10

rofile on which a structure endi et al. [BIG 12b], who showedids (one in tension and one inicture. This structure, as shownan additional elastic spring on1.4.

loth and described in the X1-X2

1] and f'(O) = O.

~ a deformed mode defined bykwise. The potential energy of


Figure 1.3. Post-critical behavior in tension ofa single-degree-of-freedomstructure . The structure has two critical loads, one in tension and one in

compression [BIG 12b]

:in ¢)] , [1.4]

When the profile of the constraint is circular, with radius R; and dimensionlesssigned curvature X= 1'1/[1 + (1')2]3/2 = ±l/R; as shown in the inset of Figures 1.5and 1.6, the axial load at equilibrium satisfies

[1.5]F = _ k1¢)I- X2 sin

2¢ +

l sin¢(x cos¢ + )1 - X2 sin 2 ¢)

k2 [¢ + sin-1 (X sin ¢) - 7r H(X¢ )]

l sin ¢

[1.6]


where H denotes the Heaviside step function . Since (3(cP) = - tan-1 [I' (sin cP )] - cP,the critical load of the system is

Equation [1.10] has been t

£ is discontinuous at cP = 0), 1

k 1 + k2 [1 + f"(0)]2l[l + f"(O)]

[1.7]

shorte ning ,

0.50

where f "(O) = £(0) is the signed curvature at cP = O.0.25

,Y.

':Ii.<

0.00

-0.25

-0.50

-2

<1>0= 0°, perf,;;---:::::=:=~::::.._ .

-~

-~--

-1

Figure 1.4. A single-degree-of-freedom structure with a linear-ela stic hingeconstrained to slide along a generic profile at the right end and a rotational

linear-elastic spring at the left end

Figure 1.5. The ~

single -degree-of-freedonand k2/k1 = 0.01, evide

tensile. Note

For an imperfect system, characterized by an initial inclination of the rod cPo, thepotential energy becomes

shortenin g,

-1

c-::1f-~~' _-::, ,6ii, ,

-2

10.00

-10.00

5.00

~':-Ii.<

0.00

-5 .00

[1.8]

[1.9]k 1 (cP - cPo) + k2 [(3 (cP) - (3 (cPo) ] (3' (cP)

l [sin cP + cos cP f' (sin cP) ]

- F l [cos cP - f(sin cP) - cos cPo+ f (sin cPo) ] , <1>0=0°, perf.

F=

so that the axial force at equilibrium is

which for a circular profile becomes

F=- k1(¢- ¢oh/1 - x2sin2 ¢ +l sin ¢( x cos¢ + )1- X2 sin2 ¢)

k2 [¢ - ¢o + sin-1(x sin¢) + sign(x¢)sin - 1(x sin ¢o) - 7fH(X¢)]l sin¢

[1.10]

Figure 1.6. The ,single-degree -of-freedom sk2/k1 = 0.5, evidencing

Note that at points label

:¢) = -tan-1[j'(sin¢)] - ¢,


Equation [1.10] has been used for X= ±4, with an "S-shaped" constraint (so thatXis discontinuous at ¢ = 0), to obtain the results shown in Figures 1.5 and 1.6.

shortening, 8<0 length ening, 8>0

[1.7]0.50

0.25

~

""~ 0.00

-0.25

-0.50

-2

l\kl l LY\F-;~ -r- .

3

_~ bim,cation,~~

-~-~: -~

-1

.::. ~Q)

0Soo

th a linear-elastic hingeght end and a rotationalend

Figure 1.5. The force versus end displacement behavior ofasingle-degree-of-freedom structure, with an "Srshaped" constraint, X= ±4and k2/k1 = 0.01, evidencing two buckling loads, one compressive and one

tensile. Note the four points where the force vanishes

I inclination of the rod ¢o, the

+[1.8]

lin¢o)] ,

[1.9]

10.00

5.00

~-

""~ 0.00

-5.00

-10.00

-2

short ening , 8<0

l\k, J LY\F-;~" " ' ''~ ' ' ' ' '' ' '

<l>0=-80~<1>0= -5°

<1>0=0 °, perfect system

-1

lengthening, 8>0

bifurcati on, 1.83

/1;<1>0= 0: , perfect syste m<1>0= 5

1 <1>0= 8° 2

[1.10]

;in¢o) - 7rH(X¢)]

Figure 1.6. The force versus end displacement behavior ofasingle-degree-of-freedom structure, with an "S-shaped" constraint, X= ±4 andk2/k1 = 0.5, evidencing two buckling loads, one compressive and one tensile.

Note that at points labelled "2" and "5 ", the externalforce does not vanish


1.4. The Ziegler pendulum made unstable by Coulomb friction

The first experimental evidence of flutter and divergence instability related to dryfriction has recently been provided by Bigoni and Noselli [BIG 11]. In theirexperimental study, essentially based on the Ziegler's double pendulum [ZIE 77],Coulomb friction was exploited in order to provide the system with a tangentialfollower force of frictional origin. This goal was achieved by endowing the doublependulum with a freely rotating wheel, constrained to slide with friction on ahorizontal plate (see Figure 1.7 for the experimental setting and Figure 1.8 for asequence of images revealing flutter instability) .

Figure 1.7. The experimental setting used by Bigoni and Noselli [BIG 11] to show theconnection between Coulomb friction and dynamic instabilities such as flutter and divergence.

A Ziegler double pendulum is endowed at its tip with a freely rotating wheel, constrained toslide on a horizontal plate and providing the system with a follower force offrictional origin

Note that, to generate a force of the frictional type, a transversal reaction betweenplate and wheel is needed, which during the experiments was created by hanging adead weight W on the left of the structure, used as a lever.

In their experimental study, Bigoni and Noselli [BIG 11] analyzed the stability ofthe double pendulum using the five different wheels, as shown in Figure 1.9; however,in their numerical analyses, the wheel was assumed to be massless, so the aim of thissection is to show the effects on the system's dynamic of a heavy wheel.

Figure 1.8. A sequence qhandycam at 25 frames p,exhibiting flutter instabil

0.40 s and the til

Figure 1.9. The five differentNoselli [BIG 11]. (1) Aluminum vthickness 5 mm, weight 3 g; (2) (5 mm, weight 18 g; (3) cylindrical22 g; (4) steel wheel with V-shapweight 17 g; (5) cylindrical steel \1

When the mass of the whetcos a2 el +sin a2 e2) force P ,

omb friction

'gence instability related to dryd Noselli [BIG 11]. In their's double pendulum [ZIE 77],the system with a tangential

ieved by endowing the doublei to slide with friction on a1 setting and Figure 1.8 for a

!Noselli [BIG 11] to show theies such as flutter and divergence.ly rotating wheel, constrained toFollower force offrictional origin

a transversal reaction betweenents was created by hanging aver.

G 11] analyzed the stability ofshown in Figure 1.9; however,be massless, so the aim of thisof a heavy wheel.


Figure 1.8. A sequence of images (taken from a movie recorded with a Sonyhandycam at 25 frames per second) of the structure shown in Figure 1.7 andexhibiting flutter instability. The whole sequence of images was recorded in

0.40 s and the time interval between two images was 0.08 s

Figure 1.9. The five different wheels used in the experimental tests by Bigoni andNoselli [BIG 11]. (1) Aluminum wheel with V-shaped cross-section, external diameter 15 mm,thickness 5 mm, weight 3 g; (2) cylindrical steel wheel, external diameter 25 mm, thickness5 mm, weight 18 g; (3) cylindrical steel wheel, external diameter 25 mm, thickness 6 mm weight22 g; (4) steel wheel with V-shaped cross-section, external diameter 25 mm, thickness 6 mm,weight 17 g; (5) cylindrical steel wheel, external diameter 25 mm, thickness 10 mm, weight 36 g

When the mass of the wheel is taken into account, this is subject to a radial (e r =

cos a2 el +sin a2 e2) force P and to a tangential (e, = - sin a2 el +cos a2 e2) force


T applied at the contact point with the moving plate as shown in Figure 1.10. Theseforces can be expressed in component form as

The assumption of Coulomb friction at the contact point between the wheel andthe plate allows us to write

~A~I k,~ -------------- /1-------

In equation [1.14], mb m2l-i , the mass of the rod of lengand I w r are, respectively, thevertical axis and the principaland its rotation axis.

Figure 1.10. A three-degree-of-frorthogonal follower force T proviwhich moves with velocity ofmod,connected through two rotationalwheel has mass m -», radius r -» an

Now imposing condition [6a3, we arrive at the system (dynamics of the system

[1.11]

[1.13]

[1.12]if 6; = 0 ,

if 6; = 0 ,

if 6; # 0 ,

p = - P cos a 2el - P sin a 2 e2 ,

T = -Tsina2 el + Tcos a2 e2.

6; = vp cos a2 - h sin(al - a2)al ,

6; = rv» sin a2 + h cos(al - a2)al + l2a2 .

where the two scalar quantities P and T have been introduced. Note that P and T arepositive quantities when the forces acting on the wheel are directed as in Figure 1.10,and, in general, their absolute values equal to IPI and ITI, respectively.

where R is the vertical reaction applied at the wheel and orthogonal to the movingplane, J.Ls and J.Ld are the static and dynamic friction coefficients , respectively, and 6;and 6~ are the radial and the tangential components of the velocity of the wheel withrespect to the plate, which can be expressed in the forms

The system is characterized by three-degrees-of-freedom, denoted by aI, a2 anda3, and the latter representing the rotation of the wheel about its axis (see Figure 1.10).Moreover, m w , r w and hw are the mass , the radius and the thickness of the wheel.

The principle of virtual works, denoting the scalar product with " . ", is written as

p. 6C + T· (6C + rw 6a 3 e.) - (klal + ,Blad 5al +

- [k2(a2 - a l ) + ,B2(a2 - a l )](6a2 - 6al) +[1.14]

holding for every virtual displacement 6C, 6GI and 6G2, functions of the virtualrotations &~l , 6c~2 and 6a 3.


[1.11]

1S shown in Figure 1.10. These

t point between the wheel and

I k[ I I

~ -------------- I[ -------------;--------------12

- - - - - - - - - - - - -;-

Figure 1.10. A three-degree-of-freedom system subject to a tangential follower force P andorthogonal follower force T provided by afreely rotating wheel sliding withfriction on a plate,which moves with velocity ofmodulus V p . The two rods, of linear mass density p, are rigid andconnected through two rotational springs of stiffness k 1 and k2 and viscosity {31 and {32. Thewheel has mass m «, radius T w and thickness hw[1.12]. t .

Cp + CY3 T w = 0 ,

.oduced. Note that P and Tarel are directed as in Figure 1.10,TI, respectively.

and orthogonal to the movingefficients, respectively, and 6;: the velocity of the wheel with[1S

[1.13]

In equation [1.14], mI, m2 and m w are, respectively, the mass of the rod oflengthh, the mass.of the rod of length l2 and the mass of the wheel, whereas h 3 , 123 , 1w 3

and 1w r are, respectively, the principal moment of inertia of the two rods about thevertical axis and the principal moment of inertia of the wheel about the vertical axisand its rotation axis.

Now imposing condition [1.14] and invoking the arbitrariness of 8al , 8a2 and8a3, we arrive at the system of three nonlinear differential equations, governing thedynamics of the system

eedom, denoted by aI, a2 andabout its axis (see Figure 1.10).l the thickness of the wheel.

product with " . ", is written as

[1.15]

[1.14]

8G 2 , functions of the virtual


We note from equations [1.12]-[1.15] that ab a2, a3, P and T are the fiveunknowns, function of time. Moreover, in the case in which sliding between thewheel and the plate is active, a situation corresponding to 6; # 0, one additionalcondition has to be imposed in order to find the solution, namely, that the forceapplied to the wheel, P + T, must be directed parallel , but opposite to the relativeplate/wheel velocity, C+ vp el + a3 Tw et, a condition yielding

PT

[1.16]

New examples of structurebeen given, slightly generalizi:broad set of systems behaving j

and practically realized.

The effects of a constrainishown that the introduction (spring strongly affects the Pmultiple equilibrium configu:magnitude.

The nonlinear system of equations has been numerically solved, and for thispurpose the function "NDSolve" of Mathematica 6.0 has been used, together with aviscous smooth approximation of the friction law [1.12] (see [ODE 85, BIG 11]).

In Figure 1.11, a comparison is found (in terms of al and a2) between thenumerical results for the case of a massless (solid curves) and a heavy (dashed curve)-wheel. These results have been obtained for a dead weight W corresponding to theonset of flutter instability and assuming wheel number 3 as shown in Figure 1.9.From the results shown in Figure 1.11, we can conclude that the inertia of the wheelonly slightly contributes to the motion of the system and can therefore be neglected.

Finally, we have presentedby dry friction in the Zieglerrelated to a heavy frictional cthese are negligible for theexperimental setting, but may

The structures considered iand artificial materials, whichnanoscale.

Financial support from tlu286110-INTERCER2) is grate

[BIG 11] BIOONI Do, NOSELLinstabilities induced by dry j

vol. 59,noo 10,pp.2208-222(

[BIG 12a] BIOONI D., Nonlin.Instability, Cambridge Univer

[BIG 12b] BIOONI D., MISSERcurvature on structural instabiof the Royal Society A: Manoo 2144,pp. 2191-2209,201

[DOA 11] DOARE 0., MICHELIflexible plates: linear stabilitjStructures, vol. 27, no. 8, ppo

[ELI 05] ELISHAKOFF I., "ContIoverview", Applied Mechanic ,

1.7. Bibliography

1.6. Acknowledgments

542 3Time [s]

2 3Time [s]

-0.25

0.25

Figure 1.11. The instantaneous rotations 0:1 and 0:2 of the Ziegler pendulum 's rods,numerically obtained as functions oftime, for massless (solid curves) and heavy (dashed curves,the assumed wheel is number 3 in Figure 1.9) wheels. The results have been obtained fora dead load W at the onset of flutter, a plate velocity vp = 50 mm/s and initial conditions0:1 = 0:2 = 10

Instability in tension, effects of a constraint's curvature and follower loads inducedby dry Coulomb friction are new phenomena that open an important perspective in thedesign of structures that can become unstable at prescribed loads.

1.5. Conclusions

~2, a3, P and T are the fivein which sliding between theng to 6; I: 0, one additionaldution, namely, that the forceleI, but opposite to the relativen yielding

[1.16]

merically solved, and for thishas been used , together with a~] (see [ODE 85, BIG 11]).

s of a 1 and a 2) between thees) and a heavy (dashed curve)/eight W corresponding to theoer 3 as shown in Figure 1.9.de that the inertia of the wheelid can therefore.be neglected.

2 - heavy wheel

2 - massless wheel

234Time [sl

if the Ziegler pendulum's rods,curves) and heavy (dashed curves,re results have been obtained for= 50 mm/s and initial conditions

:ure and follower loads inducedan important perspective in theibed loads.


New examples of structures exhibiting buckling under tensile dead loading havebeen given, slightly generalizing previous findings by the authors and showing that abroad set of systems behaving in a counterintuitive and innovative way can be inventedand practically realized.

The effects of a constraint's curvature have been further investigated: we haveshown that the introduction on a curved constraint profile of an elastic, torsionalspring strongly affects the post-critical behavior of the system and may lead tomultiple equilibrium configurations, corresponding to an external force of zeromagnitude.

Finally, we have presented also a detailed analysis of flutter instability as inducedby dry friction in the Ziegler double pendulum. In this system, dynamical effectsrelated to a heavy frictional constraint have been determined. The results show thatthese are negligible for the values of a constraint's inertia pertinent to ourexperimental setting, but may become interesting in other situations.

The structures considered in our study can be combined to design flexible systemsand artificial materials, which may find broad applications, even at the micro- andnanoscale.

1.6. Acknowledgments

Financial support from the European FP7 - Intercer2 project (PIAP-GA-2011286110-INTERCER2) is gratefully acknowledged.

1.7. Bibliography

[BIG 11] BIGONI D ., NOSELLI G. , "Experimental evidence of flutter and divergenceinstabilities induced by dry friction" , Journal of the Mechanics and Physics of Solids ,vol. 59,no. 10,pp.2208-2226,2011.

[BIG 12a] BIGONI D ., Nonlinear Solid Mechanics. Bifurcation Theory and MaterialInstability , Cambridge University Press, 2012.

[BIG 12b] BIGONI D., MISSERONI D. , NOSELLI G. , et aI., "Effects of the constraint'scurvature on structural instability: tensile buckling and multiple bifurcations", Proceedingsof the Royal Society A: Mathematical, Physical and Engineering Sciences , vol. 468,no. 2144,pp. 2191-2209, 2012.

[DOA 11] DOARE 0 ., MICHELIN S. , "Piezoelectric coupling in energy-harvesting flutteringflexible plates: linear stability analysis and conversion efficiency", Journal of Fluids andStructures, vol. 27, no. 8, pp. 1357-1375,2011.

[ELI 05] ELISHAKOFF 1., "Controversy associated with the so-called 'follower force': criticaloverview", Applied Mechanics Reviews , vol. 58, no. 2, pp. 117-142,2005.


[KIR 10] KIRILLOV O.N., VERHULST E , "Paradoxes of dissipation-induced destabilizationor who opened Whitney's umbrella?", ZeitschriJt fUr Angewandete Mathematik undMechanik, vol. 90,no.6,pp.462-488,2010.

[KGI96] KOlTER W.T. , "Unrealistic follower forces", Journal of Sound and Vibration,vol. 194, no. 4,pp. 636-638, 1996.

[ODE 85] ODEN J.T. , MARTINS J.A.C., "Models and computational methods for dynamicfriction phenomena", Computer Methods in Applied Mechanics and Engineering, vol. 52,no. 1-3,pp.527-634, 1985.

[SUG 95] SUGIYAMA Y, KATAYAMA K. , KINO! S., "Flutter of a cantilevered column underrocket thrust", Journal ofAerospace Engineering, vol. 8, no. 1, pp. 9-15, 1995.

[ZAC 11] ZACCARIA D., BIGONI D., NOSELLI G. , et al., "Structures buckling under tensiledead load", Proceedings of the Royal Society A: Mathematical, Physical and EngineeringSciences , vol. 467 , no. 2130 , pp. 1686-1700,2011.

[ZIE 77] ZIEGLER H ., Principles of Structural Stability, Birkhauser Verlag, Basel, Stuttgart,1977.

WKB SobEquilibr

In this chapter, we presentproblems for the Schrodingerobtained in a series of papers Iproperties of the underlying cl:have studied the case where thHamiltonian flow. This occurs,Much is encoded in what wewhich we describe here in detaiCauchy problem is that of thepoint.

2.1. Introduction

In this chapter, we sum up I

BON 11, BON, ALE 08] conesemi-classical Schrodinger ope

P = -h2!1 +V(x) ,

Chapter written by Jean-Fran.Maher ZERZERI.

spectralanalysis, stability andbifurcations systems nonlinear...

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