flexural-torsional flutter and buckling of braced foil...

Research ArticleFlexural-Torsional Flutter and Buckling of Braced Foil Beamsunder a Follower Force

Manuel Ferretti12 Francesco DrsquoAnnibale12 and Angelo Luongo12

1 International Research Center on Mathematics and Mechanics of Complex Systems University of LrsquoAquila 67100 LrsquoAquila Italy2Department of Civil Construction-Architectural and Environmental Engineering University of LrsquoAquila 67100 LrsquoAquila Italy

Correspondence should be addressed to Angelo Luongo angeloluongounivaqit

Received 30 May 2017 Revised 13 August 2017 Accepted 24 August 2017 Published 12 October 2017

Academic Editor Salvatore Caddemi

Copyright copy 2017 Manuel Ferretti et alThis is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

The flutter and buckling behavior of a cantilever foil beam loaded at the tip by a follower force are addressed in this paper Thebeam is internally and externally damped and braced at the tip by a linear spring-damper device which is located in an eccentricposition with respect to beam axis thus coupling the flexural and torsional behaviors An exact linear stability analysis is carriedout and the linear stability diagram of the trivial rectilinear configuration is built up in the space of the follower load and springrsquosstiffness parameters The effects of the flexural-torsional coupling as well as of the damping on the flutter and buckling criticalloads are discussed

1 Introduction

Dynamic stability of elastic systems loaded by nonconser-vative and configuration-dependent loads such as followerforces [1 2] has been thoroughly investigated by manyresearchers in the last century [3ndash10] Some experimentalevidences proving the existence of such a kind of actionsin the real world applications are available for example in[10ndash12] and in the critical review [13] notwithstanding thefact that the engineering world is still suspicious of theirexistence and physical meaning However the effects ondynamic stability due to the presence of follower forces arevery important in several engineering branches such as inaerospace [10 14 15] in flexible pipes conveying fluid [16ndash18]and in vehicle brakes [19 20]

Researchers have devoted great attention in the last yearsto the so-called Beckrsquos beam (see eg [4 5 13]) namely acantilever beam loaded at the tip by a follower force (ie aforce which keeps its direction tangential to the centerline)and eventually in the presence of conservative loads andorof distributed (internal and external) as well as lumped formsof damping This structure indeed represents a paradigmaticsystem for the comprehension of stability issues in one-dimensional nonconservative systems in fact the loss ofstability may happen either by divergence in the presence

of conservative loads or lumped springs or by flutter alsosaid to be Hopf bifurcation in Dynamical System Theorydepending on the mechanical properties of the structure [21ndash24]Moreover Beckrsquos beam is also able to showone of themostamazing phenomena occurring in the dynamical behaviorof elastic systems loaded by follower forces namely thedestabilizing effect of damping or the ldquoZiegler Paradoxrdquo seefor example [3ndash6 25ndash27] It occurs when a vanishingly smalland positive-definite damping is added to such a systementailing a finite reduction of the flutter critical load withrespect to that of the undamped system Several contributionscan be found in the literature which are devoted to giving anexplanation of this occurrence and to present different casestudies pointing out the phenomenon among the others thereader can refer to [8 9 24 28ndash33]

In most of the previously cited papers a planar Beckrsquosbeam under in-plane loads is considered whose trivialrectilinear configuration loses its stability in the same planeHowever when considering spatial beams the loss of stabilitycan occur out of that plane due to a flexural-torsionalmechanical coupling This phenomenon is well known inbuckling analysis of spatial structures such as thin-walledmembers which indeed can exhibit a flexural-torsionalEulerian bifurcation when subjected to conservative forces[34 35]

HindawiMathematical Problems in EngineeringVolume 2017 Article ID 2691963 10 pageshttpsdoiorg10115520172691963

2 Mathematical Problems in Engineering

The flexural-torsional coupling may become importantwhen issues relevant to dynamic stability are addressedIn this framework classical examples can be found mainlyin aerospace engineering for example when the flutterbehavior of a wing immersed in a gas flow namely undernonconservative and velocity-dependent loads is considered[5 36] Other examples when configuration-dependent loadsact can be found in [37] where the flutter instability ofa cantilever beam containing a tip mass subjected to atransverse follower force at the tip and in the presenceof airflow is addressed in [38] where the lateral-torsionalstability of deep cantilever beams loaded by a transversefollower force at the tip is studied in [39] where the lateralstability of a slender beam under a transverse follower forceis addressed in [21] where the flexural-torsional bifurcationsof a cantilever beam under the simultaneous action of anonconservative follower force and a conservative couple atthe free end have been analyzed andfinally in [40] where thebending-torsional flutter analysis of a cantilever containingan arbitrarily placed mass under a follower force and airflowis analyzed Remarkably in the greatest part of the previouspapers a foil beam namely a beam for which one of the twoinertia moments is much larger than the other is consideredas the mathematical model of aircraftrsquos wing

This paper is framed in the scenario illustrated aboveIndeed to the best of authorrsquos knowledge there are nocontributions in the literature addressing the flutter andbuckling analyses of a spatial Beckrsquos column so that thepresent work is a first step toward the study of the problemTo this end reference will be made to the simplest modelas possible namely a clamped-free foil beam loaded at thetip by a tangential follower force internally and externallydamped Moreover in order to couple the flexural andtorsional behavior even in the linear range it is assumed thatthe beam is braced at the tip by a linear spring-damper devicewhich is orthogonal to the axis line and eccentricwith respectto it

The paper is organized as follows In Section 2 theequations of motion of the model are presented In Section 3the eigenvalue problem is addressed and an exact linearstability analysis is carried out both in the presence and inthe absence of damping In Section 4 a numerical analysis isdeveloped and the linear stability diagrams are built up in thespace of the follower load and springrsquos stiffness parametersand for different damping coefficients Finally in Section 5some conclusions are drawn

2 Model

The foil beam is modeled as one-dimensional inextensibleand twistable polar continuum (see eg [41]) embeddedin a three-dimensional space spanned by the unit vectorsa119909 a119910 and a119911 (Figure 1) It is assumed that 119911 is the strongand 119910 the weak axis of the cross-section that is the inertiamoments are 119869119911 ≫ 119869119910 consistently the 119909119910-plane is herereferred to as the strong plane while 119909119911-plane is the weakone If forces in the strong plane are smaller or at mostcomparable with those acting in the weak plane then thebeam can be considered unflexurable in the strong plane

A

z

b

y

x

F

B

Q

h

e

k cb

ax ay

az

Figure 1 Foil beam viscoelastically braced under follower force

and the relevant bending moment 119872119911 is a reactive stressConcerning torsional stiffness it is of the same order ofmagnitude of the weak bending stiffness as it happens forcompact cross-sections Therefore if torsional moments 119872119909are smaller than the bending moment 119872119910 torsion is alsonegligible so that the foil beam behaves as a planar beamin bending Throughout the paper it is assumed that thetorsional moment is comparable with the bending momentso that the foil beam behaves as a flexural-torsional beam

The beam is clamped at 119860 119904 = 0 and free at 119861 119904 =ℓ where 119904 is the abscissa spanning the points of the axisline its flexural and torsional stiffness are 119864119869119910 and 119866119869119909respectively with 119864 and 119866 being the elastic and tangentialmoduli moreover119898 is the mass per unit length and 119868119909 is therotational moment of inertia The beam is loaded at 119861 by afollower force having intensity 119865 which keeps its directionparallel to the tangent to the centerline at 119861 The internaldissipation is assumed to be ruled by a linear Kelvin-Voigtlaw whose flexural and torsional viscosity coefficients are 120578and 120577 respectively An external viscous dissipation (eg dueto the surrounding air) is also taken into account by a uniformdistribution of linear dashpots whose flexural and torsionalviscosity coefficients are 119888119905 and 119888119903 respectively Finally thebeam is viscoelastically braced at 119861 by a linear spring-damperdevice having elastic and viscosity coefficients 119896 and 119888119887respectively it is located at the point119876 = (ℓ 119890 0) with 119890 beingits distance from the 119909-axis (see Figure 1)

By denoting by 119908 = 119908(119904 119905) the deflection in the weakplane and by 120579 = 120579(119904 119905) the twist angle 119905 being thetime the linearized equations of motion of the foil beam innondimensional form read

minus 1Λ2 120579 minus 119888119903 120579 + Γ12057910158401015840 + 120577 12057910158401015840 = 0 + 119888119905 + 1199081015840101584010158401015840 + 1205781015840101584010158401015840 + 212058311990810158401015840 = 0

(1)

in which the dot and the prime symbolize derivatives withrespect to the nondimensional time 119905 and abscissa 119904 respec-tively

Similarly the relevant boundary conditions are

120579119860 = 0119908119860 = 0

Mathematical Problems in Engineering 3

1199081015840119860 = 0Γ1205791015840119861 + 120577 120579119861 + 119896119890 (119908119861 + 119890120579119861) + 119888119887119890 (119861 + 119890 120579119861) = 0minus119908101584010158401015840119861 minus 120578101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) + 119888119887 (119861 + 119890 120579119861) = 0

11990810158401015840119861 + 12057810158401015840119861 = 0(2)

Equations (1) and (2) are obtained by introducing the follow-ing positions and by removing the tilde

= 120596119905119904 = 119904ℓ 120579 = 120579119908 = 119908ℓ 120596 = radic 119864119869119910119898ℓ4 120583 = 119865ℓ22119864119869119910 119890 = 119890ℓ 119896 = 119896ℓ3119864119869119910

Λ = radic119898ℓ2119868119909 Γ = 119866119869119909119864119869119910 119888119905 = 119888119905120596ℓ4119864119869119910 120578 = 120578120596119864 119888119903 = 119888119903120596ℓ2119864119869119910 120577 = 120577120596119869119909119864119869119910 119888119887 = 119888119887120596ℓ3119864119869119910

(3)

Finally it is important to remark that when 119890 = 0 the flexuralmotion (in linear regime) is uncoupled from the torsionalone

3 Linear Stability Analysis

Stability of the trivial equilibrium position of the beamnamely 119908 = 120579 = 0 is addressed with the aim of evaluatingthe critical value of the follower force at which flutter ordivergence bifurcations occur To this end an exact analysisof the eigenvalues of the boundary value problems for boththe undamped and damped cases is developed

31 The Flutter Load of the Undamped Foil Beam Theundamped case (119888119903 = 119888119905 = 119888119887 = 120578 = 120585 = 0) is consideredfirst The relevant problem reads

minus 1Λ2 120579 + Γ12057910158401015840 = 0 + 1199081015840101584010158401015840 + 212058311990810158401015840 = 0

(4)

together with

120579119860 = 0119908119860 = 01199081015840119860 = 0

Γ1205791015840119861 + 119896119890 (119908119861 + 119890120579119861) = 0minus119908101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) = 0

11990810158401015840119861 = 0

(5)

By letting a solution be in the form

(120579 (119904 119905)119908 (119904 119905)) = (120579 (119904)

119908 (119904)) exp (120582119905) (6)

the following boundary value problem is obtained

minus 1205822Λ2 120579 + Γ12057910158401015840 = 0 (7a)

1205822119908 + 1199081015840101584010158401015840 + 212058311990810158401015840 = 0 (7b)

120579119860 = 0 (7c)

119908119860 = 0 (7d)

1199081015840119860 = 0 (7e)

Γ1205791015840119861 + 119896119890 (119908119861 + 119890120579119861) = 0 (7f)

minus119908101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) = 0 (7g)

11990810158401015840119861 = 0 (7h)

Thefield equations (7a) and (7b) are uncoupled whichmakesthe problem easy to be solved In particular the deflection119908(119904) coincides with that of the planar Beckrsquos beam (seeeg [33]) By taking into account the geometrical boundary


conditions (7c) (7d) and (7e) the solution of the field equa-tions can be written as

120579 (119904) = 1198621120574 sin (120574119904) 119908 (119904) = 1198622 (cos (120572119904) minus cosh (120573119904))

+ 1198623 (1120572 sin (120572119904) minus 1120573 sinh (120573119904)) (8)

where 119862119895 (119895 = 1 3) are arbitrary constants and 120572 120573 and120574 are wave-numbers defined by

1205722 = 120583 + radic1205832 minus 1205822

1205732 = minus120583 + radic1205832 minus 12058221205742 = minus 1205822Λ2Γ

(9)

By replacing (8) in the mechanical boundary conditionsof problem with (7f) (7g) and (7h) and rearranging thefollowing algebraic problem is obtained

[B120579120579 B120579119908B119908120579 B119908119908

] c = 0 (10)

where c fl 119862119895119879 is the column vector of the unknownconstants and

B120579120579 fl Γ cos (120574) + 1198902119896120574 sin (120574) B120579119908 fl [119890119896 (cos (120572) minus cosh (120573)) 119890119896( sin (120572)120572 minus sinh (120573)120573 )] B119908120579 fl [119890119896120574 sin (120574) 0]119879

B119908119908 fl [[minus1205723 sin (120572) + 1205733 sinh (120573) + 119896 (cos (120572) minus cosh (120573)) 1205722 cos (120572) + 1205732 cosh (120573) + 119896( sin (120572)120572 minus sinh (120573)120573 )

minus1205722 cos (120572) minus 1205732 cosh (120573) minus120572 sin (120572) minus 120573 sinh (120573)]]

(11)

are matrices depending on the eigenvalue 120582 on the followerforce 120583 and on the stiffness and eccentricity of the spring119896 and 119890 respectively by the way of wave-numbers 120572 120573and 120574 By zeroing the determinant of the (3 times 3) matrix ofcoefficients in (10) a transcendental characteristic equation119891119906(120582 120583 119896 119890) = 0 is obtained Since the system is undampedall the eigenvalues lie on the imaginary axis in the precriticalphase that is when the value of the force is less than thecritical one namely 120582 = 119894120596

For a given set (119896 119890) the undamped critical flutter load120583119906 is found as the lowest value of the force at which apair of purely imaginary eigenvalues coalesce By taking(120583 119896) as control parameters and 119890 as an auxiliary parameter(kept fixed) this coalescence mechanism takes place on themanifoldH119906 (symbolH denoting Hopf bifurcation)

H119906 fl

119891119906 (120596 120583 119896 119890) = 0120597119891119906 (120596 120583 119896 119890)120597120596 = 0 (12)

System (12) implicitly defines a multibranch curve in the(120583 119896)-parameter plane parametrized by 120596 Only numericalsolution can be pursued for this system

32 The Flutter Load of the Damped Foil Beam When in-ternal and external (distributed and lumped) damping act

on the foil beam the linear problem is governed by the fieldequations (1) and by the boundary conditions (2) By usingthe separation of variables (6) the spatial boundary valueproblem follows

minus( 1205822Λ2 + 119888119903120582)120579 + (Γ + 120577120582) 12057910158401015840 = 0(1205822 + 119888119905120582)119908 + (1 + 120578120582)1199081015840101584010158401015840 + 212058311990810158401015840 = 0

120579119860 = 0119908119860 = 01199081015840119860 = 0

(Γ + 120577120582) 1205791015840119861 + 119890 (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0minus (1 + 120578120582)119908101584010158401015840119861 + (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0

(1 + 120578120582)11990810158401015840119861 = 0

(13)

whose solution is still given by (8) but with the wave-numbers and matrices redefined as follows


1205722 = 120583 + radic1205832 minus 120582 (119888119905 + 120582) (1 + 120578120582)1 + 120578120582

1205732 = minus120583 + radic1205832 minus 120582 (119888119905 + 120582) (1 + 120578120582)1 + 120578120582

1205742 = minus1205822Λ2 + 119888119903120582Γ + 120577120582 B120579120579 fl (Γ + 120577120582) cos (120574) + 1198902 (119896 + 119888119887120582)120574 sin (120574) B120579119908 fl [119890 (119896 + 119888119887120582) (cos (120572) minus cosh (120573)) 119890 (119896 + 119888119887120582)( sin (120572)120572 minus sinh (120573)120573 )]

B119908120579 fl [119890 (119896 + 119888119887120582)120574 sin (120574) 0]119879 B119908119908

fl [[[minus (1 + 120578120582) (1205723 sin (120572) minus 1205733 sinh (120573)) + (119896 + 119888119887120582) (cos (120572) minus cosh (120573)) (1 + 120578120582) (1205722 cos (120572) + 1205732 cosh (120573)) + (119896 + 119888119887120582)( sin (120572)120572 minus sinh (120573)120573 )

minus (1 + 120578120582) (1205722 cos (120572) + 1205732 cosh (120573)) minus (1 + 120578120582) (120572 sin (120572) + 120573 sinh (120573))]]]

(14)

To compute the (damped) flutter load 120583119889 the follow-ing algorithm is applied (i) the characteristic equation119891119889(120582 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 now depending on threeexternal and two internal damping coefficients is obtained(ii) 120582 = 120585+119894120596 is put in this equation and its real and imaginaryparts are separated thus obtaining

119891119889 (120582 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)fl 119891 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)

+ 119894119892 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0(15)

with 119891 119892 isin R (iii) 120585 = 0 is taken since at the critical flutterload a single pair of eigenvalues crosses (from the left) theimaginary axis (simple Hopf bifurcation) (iv) for a given setof parameters (119896 119890 119888119903 119888119905 119888119887 120577 120578) the system 119891 = 0 119892 = 0is solved for the two unknowns 120583 and 120596 by looking for thelowest root 120583 = 120583119889

Finally by still taking (120583 119896) as control parameters andconsidering all the remaining ones as (fixed) auxiliary param-eters the (damped) flutter mechanism takes place on themanifoldH119889

H119889 fl

119891(0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (16)

This equation implicitly defines a multibranch curve in the(120583 119896)-parameter plane parametrized by 120596 No closed-formsolutions but only numerical can be pursued for system (16)

33TheDivergence Load of the Foil Beam In order to find thedivergence boundary the locus D of the roots 120585 = 0 120596 = 0

of the characteristic equation must be found As done abovethe locus can be determined by solving the system

D fl

119891(0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (17)

It is found that the second equation of system (17) identicallyvanishes for any set of parameters while the first equationof system (17) is independent of the damping coefficients Aclosed-form solution can be found for system (17) namely

D fl 41205832 (Γ + 1198902119896) minus 2Γ119896120583 cos (radic2120583)+ Γ119896radic2120583 sin (radic2120583) = 0 (18)

This equation implicitly defines a multibranch curve in the(120583 119896)-parameter plane

4 Numerical Results

Numerical analyses are here referred to as a foil beam havinga rectangular cross-section of width 119887 and thickness ℎ (seeFigure 1) Accordingly the following (dimensional) relationshold 119869119910 = (112)119887ℎ3 119869119909 = (13)119887ℎ3 119898 fl 119887ℎ120588 and 119868119909 =(112)1198873ℎ120588 with 120588 being the mass density

An exhaustive analysis of all parameters would becumbersome Therefore the following assumptions linkingexternal and internal damping coefficients are made (i)linear distribution of local damping forces due to a uniformdisposition of external dashpots on the cross-section (ii)negligible material bulk viscous deformation Accordingly


the relevant dimensional damping coefficients satisfy thefollowing relations

119888119903 = 1121198872119888119905120577 = 13120578

(19)

Moreover a length-to-width ratio equal to 5 is considered forwhich the following nondimensional quantities read

Λ = 10radic3Γ = 21 + ]

119888119903 = 119888119905300 120577 = 43120578

(20)

with ] being the Poisson coefficient which has been takenequal to 03 In addition the chosen numerical values for thedamping coefficients are 119888119905 = 110 119888119887 = 110 and 120578 = 1100

The following case studies relevant to two differentpositions of the spring-damper device are examined

(i) case study I the foil cantilever beam is braced at thetip without eccentricity namely 119890 = 0

(ii) case study II the foil cantilever beam is braced atthe tip with the spring-damper device located themaximum distance from the 119911-axis namely 119890 = 1198872

Undamped Foil Beam The linear stability diagram of theundamped foil beam is displayed in Figure 2Here the criticalload 120583 is plotted versus the stiffness of the spring 119896 for thecase studies I (black curves) and II (gray curves) respectivelyFor each of the case studies two curves are shown in thefigure (i) D is the divergence locus at which the straightconfiguration loses stability via a static bifurcation whichas discussed above is independent of damping (ii) H119906 isthe undamped Hopf locus at which the foil beam losesstability via a dynamic bifurcation (collision of two pairs ofeigenvalues)

When 119896 = 0 the critical flutter load of the undampedBeckrsquos beam is recovered namely 120583119906 = 1002 (see eg[24 28]) When 120583 is increased from 0 it is apparent thatthe braced foil beam can exploit two different mechanismsof bifurcations depending on the magnitude of 119896 namely adynamic one (curve H119906) for small values of 119896 and a staticone (curve D) for large values of 119896 The phenomenon canbe explained by the fact that when 119896 rarr infin the springbecomes a fix support rendering the system conservativesince the transverse component of the follower forces cannotexpend any work on the transverse displacement 119908119861 = 0(see also [22] for additional references) The two curves meettangentially at a (degenerate) double-zero point at which thetwo imaginary eigenvalues of the Hopf bifurcation collide atthe zero frequency

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu

Figure 2 Linear stability diagram of the undamped foil beam inthe (120583 119896)-plane when the length-to-width-ratio is equal to 5 Blackcurves case study I (119890 = 0) Gray curves case study II (119890 = 110)H119906 undamped Hopf locus D divergence locus and DZ double-zero bifurcation point

In conclusion it is proved that the spring has a beneficialeffect on the flutter behavior of the undamped Beckrsquos beamby increasing 120583119906 up to about two times in the dynamicbifurcation range (case I) This is due to the fact that thespring increases the distance among the natural frequenciesof the unloaded beam and therefore it delays their collisioncaused by an increasing of the follower force On the contrarywhen case study II is addressed it is found that the flexural-torsional coupling due to the eccentricity of the spring isdetrimental on the dynamic bifurcation since it lowers theHopf curve as a consequence the best location for the springis 119890 = 0 However the coupling is beneficial on the staticbifurcation since now this bifurcation occurs at larger 119896-values and the (significant) lower branch of the D-locusis above that of case study I consequently the maximumeccentricity is the optimum for increasing the bifurcationstatic load

It is worth noticing that the effect of the flexural-torsionalcoupling strongly depends on the length-to-width ratioindeed the larger this ratio is the smaller the differencebetween the two case studies is In order to show thisoutcome the linear stability diagram which corresponds to alength-to-width ratio equal to 10 namelyΛ = 20radic3 is shownin Figure 3

Damped Foil Beam The linear stability diagram relevant tothe damped foil beam is displayed in Figure 4 for case studiesI (Figure 4(a)) and II (Figure 4(b)) respectively It is seen thatwhile the divergence locus is not changed by damping theHopf curves at which a generic dynamic bifurcation occursdetermined by the crossing of the imaginary axis of one pairof eigenvalues are affected by the damping parameters The


25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu


plots are obtained by considering different damping formsin the numerical simulations namely (i) lumped dashpot atthe tip (labels of curves and points marked with the apex1198891) (ii) distributed internal and external damping (labelsof curves and points marked with the apex 1198892) (iii) allthe damping forms acting simultaneously (labels of curvesand points marked with the apex 1198893) Moreover the curvesreferred to as damped beams are denoted with black lines inFigure 4while the gray ones are relevant to the undamped foilbeam

The scenario discussed in Figure 2 abruptly changeswhen a dashpot is added at the tip (changing the Hopfcurve H119906 into H1198891) Irrespectively of the magnitude of thedamping parameter 119888119887 theHopf curve is below the divergencecurve for all values of 119896 and for both the case studiesTherefore due to presence of the dashpot dynamic instabilityoccurs also for large values of 119896 The destabilizing effect ofdamping is apparent (even if just lumped at the tip) it entails120583119889 lt 120583119906 not only at 119896 = 0 but at any 119896 Remarkably theflexural-torsional coupling has not effects on this behavior

When distributed internal and external damping areconsidered (changing the Hopf curveH119906 intoH1198892) the wellknown destabilization paradox is encountered [8 9 24] Thedetrimental effect of damping on the undamped beam isconfirmed at small values of 119896 although it depends on theratio between the damping parameters 119888119905 and 120578 it is worthnoticing that an interval of values of 119896 can in principle existfor which distributed damping produces instead a beneficialeffect Again the flexural-torsional coupling does not changethis behavior

It is important to remark that the destabilization phe-nomenon is triggered in Beckrsquos column by internal damping

[5] that is by a form of dissipation depending on theviscoelastic (Kelvin-Voigt) properties of the column whichis indeed proportional to the stiffness distribution externaldamping that is dissipation proportional to the mass distri-bution is instead stabilizing It is worth noticing that Beckrsquoscolumn is to be considered a particular case in this contextsince in general elastic systems loaded by nonconservativeand configuration-dependent loads stabilizing damping issophisticatedly related to both the mass and stiffness distri-bution [42] as amatter of fact while the air drag is stabilizingfor Beckrsquos column it is instead destabilizing for Pflugerrsquoscolumn [43] namely a Beckrsquos beam with a different massdistribution that is with an added point mass at the loadedend

When however a dashpot is added at the tip of aninternally and externally damped beam (changing the Hopfcurve H1198892 into H1198893) it has a beneficial effect when 119896 issmall by partially counteracting the destabilization paradoxnotwithstanding the fact that it is not sufficient to bringthe load at 120583119906 when 119896 = 0 When 119896 is sufficiently largethis beneficial effect ends and the external dashpot becomesdetrimental since it rendersH1198893 lower thanH1198892 As beforethe flexural-torsional coupling is not able to qualitativelychange the behavior of the damped planar beam

Finally the critical manifold defined by (16) which isa hypersurface in the (120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)-parameter spacecan be conveniently represented by taking (120578 119888119905) as controlparameters and the remaining ones as (fixed) auxiliaryparameters and by performing sections at 120583 = const Thecorrespondent 120583-isolines of the foil beamrsquos critical manifoldwhen 119888119887 = 0 are displayed in Figure 5 for the unbracedbeam (continuous black curves) for case study I (dashedblack curves) and for case study II (continuous gray curves)respectively It is found that the presence of the springenlarges the stable region since it moves to the left of the120583-isolines of the unbraced beam However for the selectedvalue of 119896 namely 119896 = 1 there is no significant differencebetween the two case studies as it is also confirmed by thedamped Hopf curves in Figures 4(a) and 4(b) which indeedare nearly coincident when 119896 is small

5 Conclusions

Theflutter and buckling behavior of a 3D cantilever foil beamloaded at the tip by a follower force internally and externallydamped have been investigated in this paper The role ofdifferent forms of damping distributed and lumped as wellas of the flexural-torsional coupling has been explored Thelatter has been triggered by a linear spring-damper devicelocated in an eccentric position with respect to beam axis

The linearized equations of motion of the system havebeen recalled Then the relevant linear stability diagramshave been built up via an exact analysis of the eigenvalues ofthe associated boundary value problem

In particular the following conclusions can be drawn

(1) The flexural-torsional coupling has a destabilizingeffect on the dynamic stability of both damped andundamped beams


25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)

Figure 4 Linear stability diagrams of the damped foil beam in the (120583 119896)-plane (a) case study I (b) case study II Black curves damped beamGray curves undamped beamH119906 undamped Hopf locusH1198891 Hopf locus for lumped dashpot at the tipH1198892 Hopf locus for distributedinternal and external dampingH1198893 Hopf locus for all the damping forms acting simultaneouslyD divergence locus and DZ double-zerobifurcation point

010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002

Figure 5 120583-isolines in the (120578 119888119905)-plane when 119896 = 1 Continuousblack curves unbraced beam Dashed black curves case study IContinuous gray curves case study II U unstable region S stableregion

(2) The flexural-torsional coupling is stabilizing for thestatic bifurcation

(3) The stabilizing and destabilizing effects of the flexu-ral-torsional coupling are reduced when the length-to-width-ratio is increased

(4) The lumped dashpot at the tip has a detrimental effecton the dynamic stability this effect does not dependon the flexural-torsional coupling that is it holds forany stiffness and eccentricity of the lumped spring-damper device

(5) The effect of coupling does not qualitatively changethe dynamic stability of damped beams

Conflicts of Interest

The authors declare that there are no conflicts of interestregarding the publication of this paper

References

[1] W Koiter ldquoUnrealistic follower forcesrdquo Journal of Sound andVibration vol 194 no 4 p 636 1996

[2] Y SugiyamaM A Langthjem and B-J Ryu ldquoRealistic followerforcesrdquo Journal of Sound and Vibration vol 225 no 4 pp 779ndash782 1999

[3] H Ziegler ldquoDie Stabilitatskriterien der ElastomechanikrdquoIngenieur-Archiv vol 20 no 1 pp 49ndash56 1952

[4] M Beck ldquoDie Knicklast des einseitig eingespannten tangentialgedruckten Stabesrdquo ZAMP Zeitschrift fur angewandte Mathe-matik und Physik vol 3 no 3 pp 225ndash228 1952

[5] V V Bolotin Nonconservative Problems of the Theory of ElasticStability Corrected and authorized edition Translated fromthe Russian by T K Lusher English translation edited by GHerrmann A Pergamon Press Book The Macmillan Co NewYork 1963


[6] G Herrmann ldquoStability of equilibrium of elastic systems sub-jected to non-conservative forcesrdquo Applied Mechanics Reviewsvol 20 pp 103ndash108 1967

[7] I P Andreichikov and V I Yudovich ldquoThe stability of visco-elastic rodsrdquo Izvestiya Akad Nauk SSSR Mekhanika TverdogoTela vol 9 no 2 pp 78ndash87 1974

[8] A P Seyranian and A A Mailybaev Multiparameter StabilityTheory with Mechanical Applications vol 13 of Series on Sta-bility Vibration and Control of Systems Series A TextbooksMonographs and Treatises World Scientific Publishing Co IncSingapore Singapore 2003

[9] O N Kirillov Nonconservative stability problems of modernphysics vol 14 of De Gruyter Studies in Mathematical PhysicsDe Gruyter Berlin 2013

[10] M A Langthjem and Y Sugiyama ldquoDynamic stability ofcolumns subjected to follower loads a surveyrdquo Journal of Soundand Vibration vol 238 no 5 pp 809ndash851 2000

[11] K Ingerle ldquoStability of massless non-conservative elastic sys-temsrdquo Journal of Sound and Vibration vol 332 no 19 pp 4529ndash4540 2013

[12] D Bigoni and G Noselli ldquoExperimental evidence of flutter anddivergence instabilities induced by dry frictionrdquo Journal of theMechanics and Physics of Solids vol 59 no 10 pp 2208ndash22262011

[13] I Elishakoff ldquoControversy associated with the so-calledlsquofollower forcesrsquo critical overviewrdquo Applied Mechanics Reviewsvol 58 no 2 pp 117ndash142 2005

[14] B J Ryu and Y Sugiyama ldquoDynamic stability of cantileveredtimoshenko columns subjected to a rocket thrustrdquo Computersand Structures vol 51 no 4 pp 331ndash335 1994

[15] AMazidi S A Fazelzadeh and PMarzocca ldquoFlutter of aircraftwings carrying a powered engine under roll maneuverrdquo Journalof Aircraft vol 48 no 3 pp 874ndash883 2011

[16] H Troger and A Steindl Nonlinear stability and bifurcationtheory Springer-Verlag Vienna 1991

[17] M P Paıdoussis and N T Issid ldquoDynamic stability of pipesconveying fluidrdquo Journal of Sound and Vibration vol 33 no 3pp 267ndash294 1974

[18] L Wang ldquoFlutter instability of supported pipes conveying fluidsubjected to distributed follower forcesrdquo Acta Mechanica SolidaSinica vol 25 no 1 pp 46ndash52 2012

[19] J E Mottershead ldquoVibration- and friction-induced instabilityin disksrdquo Shock and Vibration Digest vol 30 no 1 pp 14ndash311998

[20] N M Kinkaid O M OrsquoReilly and P Papadopoulos ldquoAutomo-tive disc brake squealrdquo Journal of Sound and Vibration vol 267no 1 pp 105ndash166 2003

[21] A Paolone M Vasta and A Luongo ldquoFlexural-torsional bifur-cations of a cantilever beam under potential and circulatoryforces I Non-linear model and stability analysisrdquo InternationalJournal of Non-Linear Mechanics vol 41 no 4 pp 586ndash5942006

[22] A Di Egidio A Luongo and A Paolone ldquoLinear and non-linear interactions between static and dynamic bifurcationsof damped planar beamsrdquo International Journal of Non-LinearMechanics vol 42 no 1 pp 88ndash98 2007

[23] A Luongo and F DrsquoAnnibale ldquoDouble zero bifurcationof non-linear viscoelastic beams under conservative andnon-conservative loadsrdquo International Journal of Non-LinearMechanics vol 55 pp 128ndash139 2013

[24] A Luongo and F DrsquoAnnibale ldquoOn the destabilizing effectof damping on discrete and continuous circulatory systemsrdquo

Journal of Sound and Vibration vol 333 no 24 pp 6723ndash67412014

[25] H Leipholz ldquoUber den Einfluszlig der Dampfung bei nichtkon-servativen Stabilitatsproblemen elastischer Staberdquo Ingenieur-Archiv vol 33 no 5 pp 308ndash321 1964

[26] R H Plaut ldquoA New Destabilization Phenomenon in Noncon-servative SystemsrdquoZAMM- Journal of AppliedMathematics andMechanics vol 51 no 4 pp 319ndash321 1971

[27] P Hagedorn ldquoOn the destabilizing effect of non-linear dampingin non-conservative systems with follower forcesrdquo InternationalJournal of Non-LinearMechanics vol 5 no 2 pp 341ndash358 1970

[28] O N Kirillov and A P Seyranian ldquoThe effect of small internaland external friction on the stability of distributed nonconser-vative systemsrdquo Journal of Applied Mathematics and Mechanicsvol 69 no 4 pp 584ndash611 2005

[29] O N Kirillov and F Verhulst ldquoParadoxes of dissipation-induced destabilization or who opened Whitneyrsquos umbrellardquoZAMM Zeitschrift fur Angewandte Mathematik und Mechanikvol 90 no 6 pp 462ndash488 2010

[30] Y G Panovko and S V Sorokin ldquoQuasi-stability of viscoelasticsystems with tracking forcesrdquoMechanics of solids vol 22 no 5pp 128ndash132 1987

[31] N Zhinzher ldquoInfluence of dissipative forces with incompletedissipation on the stability of elastic systemsrdquo Mechanics ofSolids vol 29 no 1 pp 135ndash141 1994

[32] A Luongo and F DrsquoAnnibale ldquoA paradigmatic minimal systemto explain the Ziegler paradoxrdquo Continuum Mechanics andThermodynamics vol 27 no 1-2 pp 211ndash222 2015

[33] F DrsquoAnnibaleM Ferretti andA Luongo ldquoImproving the linearstability of the Beckrsquos beam by added dashpotsrdquo InternationalJournal of Mechanical Sciences vol 110 pp 151ndash159 2016

[34] M Pignataro N Rizzi and A Luongo Stability and PostcriticalBehaviour of Elastic Structures Elsevier Science PublishersAmsterdam Netherlands 1990

[35] N S Trahair Flexural-Torsional Buckling of Structures vol 6CRC Press Boca Raton Fla USA 1993

[36] D H Hodges and G A Pierce ldquoIntroduction to structuraldynamics and aeroelasticity second editionrdquo Introduction toStructural Dynamics and Aeroelasticity Second Edition pp 1ndash247 2011

[37] W T Feldt and G Herrmann ldquoBending-torsional flutter ofa cantilevered wing containing a tip mass and subjected to atransverse follower forcerdquo Journal of the Franklin Institute vol297 no 6 pp 467ndash478 1974

[38] D H Hodges ldquoLateral-torsional flutter of a deep cantileverloaded by a lateral follower force at the tiprdquo Journal of Soundand Vibration vol 247 no 1 pp 175ndash183 2001

[39] F M Detinko ldquoSome phenomena for lateral flutter of beamsunder follower loadrdquo International Journal of Solids and Struc-tures vol 39 no 2 pp 341ndash350 2001

[40] S A Fazelzadeh A Mazidi and H Kalantari ldquoBending-torsional flutter of wings with an attached mass subjected to afollower forcerdquo Journal of Sound and Vibration vol 323 no 1-2pp 148ndash162 2009

[41] A Luongo and D Zulli Mathematical Models of Beams andCables John Wiley Sons Inc Hoboken NJ USA 2013

[42] ONKirillov andA P Seyranian ldquoStabilization anddestabiliza-tion of a circulatory system by small velocity-dependent forcesrdquo


Journal of Sound and Vibration vol 283 no 3-5 pp 781ndash8002005

[43] M Tommasini O N Kirillov D Misseroni and D BigonildquoThe destabilizing effect of external damping Singular flutterboundary for the Pfluger column with vanishing externaldissipationrdquo Journal of the Mechanics and Physics of Solids vol91 pp 204ndash215 2016

Submit your manuscripts athttpswwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Journal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


The flexural-torsional coupling may become importantwhen issues relevant to dynamic stability are addressedIn this framework classical examples can be found mainlyin aerospace engineering for example when the flutterbehavior of a wing immersed in a gas flow namely undernonconservative and velocity-dependent loads is considered[5 36] Other examples when configuration-dependent loadsact can be found in [37] where the flutter instability ofa cantilever beam containing a tip mass subjected to atransverse follower force at the tip and in the presenceof airflow is addressed in [38] where the lateral-torsionalstability of deep cantilever beams loaded by a transversefollower force at the tip is studied in [39] where the lateralstability of a slender beam under a transverse follower forceis addressed in [21] where the flexural-torsional bifurcationsof a cantilever beam under the simultaneous action of anonconservative follower force and a conservative couple atthe free end have been analyzed andfinally in [40] where thebending-torsional flutter analysis of a cantilever containingan arbitrarily placed mass under a follower force and airflowis analyzed Remarkably in the greatest part of the previouspapers a foil beam namely a beam for which one of the twoinertia moments is much larger than the other is consideredas the mathematical model of aircraftrsquos wing

This paper is framed in the scenario illustrated aboveIndeed to the best of authorrsquos knowledge there are nocontributions in the literature addressing the flutter andbuckling analyses of a spatial Beckrsquos column so that thepresent work is a first step toward the study of the problemTo this end reference will be made to the simplest modelas possible namely a clamped-free foil beam loaded at thetip by a tangential follower force internally and externallydamped Moreover in order to couple the flexural andtorsional behavior even in the linear range it is assumed thatthe beam is braced at the tip by a linear spring-damper devicewhich is orthogonal to the axis line and eccentricwith respectto it

The paper is organized as follows In Section 2 theequations of motion of the model are presented In Section 3the eigenvalue problem is addressed and an exact linearstability analysis is carried out both in the presence and inthe absence of damping In Section 4 a numerical analysis isdeveloped and the linear stability diagrams are built up in thespace of the follower load and springrsquos stiffness parametersand for different damping coefficients Finally in Section 5some conclusions are drawn

2 Model

The foil beam is modeled as one-dimensional inextensibleand twistable polar continuum (see eg [41]) embeddedin a three-dimensional space spanned by the unit vectorsa119909 a119910 and a119911 (Figure 1) It is assumed that 119911 is the strongand 119910 the weak axis of the cross-section that is the inertiamoments are 119869119911 ≫ 119869119910 consistently the 119909119910-plane is herereferred to as the strong plane while 119909119911-plane is the weakone If forces in the strong plane are smaller or at mostcomparable with those acting in the weak plane then thebeam can be considered unflexurable in the strong plane

A

z

b

y

x

F

B

Q

h

e

k cb

ax ay

az

Figure 1 Foil beam viscoelastically braced under follower force

and the relevant bending moment 119872119911 is a reactive stressConcerning torsional stiffness it is of the same order ofmagnitude of the weak bending stiffness as it happens forcompact cross-sections Therefore if torsional moments 119872119909are smaller than the bending moment 119872119910 torsion is alsonegligible so that the foil beam behaves as a planar beamin bending Throughout the paper it is assumed that thetorsional moment is comparable with the bending momentso that the foil beam behaves as a flexural-torsional beam

The beam is clamped at 119860 119904 = 0 and free at 119861 119904 =ℓ where 119904 is the abscissa spanning the points of the axisline its flexural and torsional stiffness are 119864119869119910 and 119866119869119909respectively with 119864 and 119866 being the elastic and tangentialmoduli moreover119898 is the mass per unit length and 119868119909 is therotational moment of inertia The beam is loaded at 119861 by afollower force having intensity 119865 which keeps its directionparallel to the tangent to the centerline at 119861 The internaldissipation is assumed to be ruled by a linear Kelvin-Voigtlaw whose flexural and torsional viscosity coefficients are 120578and 120577 respectively An external viscous dissipation (eg dueto the surrounding air) is also taken into account by a uniformdistribution of linear dashpots whose flexural and torsionalviscosity coefficients are 119888119905 and 119888119903 respectively Finally thebeam is viscoelastically braced at 119861 by a linear spring-damperdevice having elastic and viscosity coefficients 119896 and 119888119887respectively it is located at the point119876 = (ℓ 119890 0) with 119890 beingits distance from the 119909-axis (see Figure 1)

By denoting by 119908 = 119908(119904 119905) the deflection in the weakplane and by 120579 = 120579(119904 119905) the twist angle 119905 being thetime the linearized equations of motion of the foil beam innondimensional form read

minus 1Λ2 120579 minus 119888119903 120579 + Γ12057910158401015840 + 120577 12057910158401015840 = 0 + 119888119905 + 1199081015840101584010158401015840 + 1205781015840101584010158401015840 + 212058311990810158401015840 = 0

(1)

in which the dot and the prime symbolize derivatives withrespect to the nondimensional time 119905 and abscissa 119904 respec-tively

Similarly the relevant boundary conditions are

120579119860 = 0119908119860 = 0


1199081015840119860 = 0Γ1205791015840119861 + 120577 120579119861 + 119896119890 (119908119861 + 119890120579119861) + 119888119887119890 (119861 + 119890 120579119861) = 0minus119908101584010158401015840119861 minus 120578101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) + 119888119887 (119861 + 119890 120579119861) = 0

11990810158401015840119861 + 12057810158401015840119861 = 0(2)


= 120596119905119904 = 119904ℓ 120579 = 120579119908 = 119908ℓ 120596 = radic 119864119869119910119898ℓ4 120583 = 119865ℓ22119864119869119910 119890 = 119890ℓ 119896 = 119896ℓ3119864119869119910

Λ = radic119898ℓ2119868119909 Γ = 119866119869119909119864119869119910 119888119905 = 119888119905120596ℓ4119864119869119910 120578 = 120578120596119864 119888119903 = 119888119903120596ℓ2119864119869119910 120577 = 120577120596119869119909119864119869119910 119888119887 = 119888119887120596ℓ3119864119869119910

(3)





minus 1Λ2 120579 + Γ12057910158401015840 = 0 + 1199081015840101584010158401015840 + 212058311990810158401015840 = 0

(4)

together with

120579119860 = 0119908119860 = 01199081015840119860 = 0

Γ1205791015840119861 + 119896119890 (119908119861 + 119890120579119861) = 0minus119908101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) = 0

11990810158401015840119861 = 0

(5)


(120579 (119904 119905)119908 (119904 119905)) = (120579 (119904)

119908 (119904)) exp (120582119905) (6)


minus 1205822Λ2 120579 + Γ12057910158401015840 = 0 (7a)

1205822119908 + 1199081015840101584010158401015840 + 212058311990810158401015840 = 0 (7b)

120579119860 = 0 (7c)

119908119860 = 0 (7d)

1199081015840119860 = 0 (7e)

Γ1205791015840119861 + 119896119890 (119908119861 + 119890120579119861) = 0 (7f)

minus119908101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) = 0 (7g)

11990810158401015840119861 = 0 (7h)





+ 1198623 (1120572 sin (120572119904) minus 1120573 sinh (120573119904)) (8)


1205722 = 120583 + radic1205832 minus 1205822


(9)


[B120579120579 B120579119908B119908120579 B119908119908

] c = 0 (10)





(11)



H119906 fl

119891119906 (120596 120583 119896 119890) = 0120597119891119906 (120596 120583 119896 119890)120597120596 = 0 (12)




minus( 1205822Λ2 + 119888119903120582)120579 + (Γ + 120577120582) 12057910158401015840 = 0(1205822 + 119888119905120582)119908 + (1 + 120578120582)1199081015840101584010158401015840 + 212058311990810158401015840 = 0

120579119860 = 0119908119860 = 01199081015840119860 = 0

(Γ + 120577120582) 1205791015840119861 + 119890 (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0minus (1 + 120578120582)119908101584010158401015840119861 + (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0

(1 + 120578120582)11990810158401015840119861 = 0

(13)



1205722 = 120583 + radic1205832 minus 120582 (119888119905 + 120582) (1 + 120578120582)1 + 120578120582



B119908120579 fl [119890 (119896 + 119888119887120582)120574 sin (120574) 0]119879 B119908119908



(14)


119891119889 (120582 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)fl 119891 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)

+ 119894119892 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0(15)



H119889 fl

119891(0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (16)




D fl

119891(0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (17)




4 Numerical Results





119888119903 = 1121198872119888119905120577 = 13120578

(19)


Λ = 10radic3Γ = 21 + ]

119888119903 = 119888119905300 120577 = 43120578

(20)







25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu









5 Conclusions






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)


010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002








References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





1199081015840119860 = 0Γ1205791015840119861 + 120577 120579119861 + 119896119890 (119908119861 + 119890120579119861) + 119888119887119890 (119861 + 119890 120579119861) = 0minus119908101584010158401015840119861 minus 120578101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) + 119888119887 (119861 + 119890 120579119861) = 0

11990810158401015840119861 + 12057810158401015840119861 = 0(2)


= 120596119905119904 = 119904ℓ 120579 = 120579119908 = 119908ℓ 120596 = radic 119864119869119910119898ℓ4 120583 = 119865ℓ22119864119869119910 119890 = 119890ℓ 119896 = 119896ℓ3119864119869119910

Λ = radic119898ℓ2119868119909 Γ = 119866119869119909119864119869119910 119888119905 = 119888119905120596ℓ4119864119869119910 120578 = 120578120596119864 119888119903 = 119888119903120596ℓ2119864119869119910 120577 = 120577120596119869119909119864119869119910 119888119887 = 119888119887120596ℓ3119864119869119910

(3)





minus 1Λ2 120579 + Γ12057910158401015840 = 0 + 1199081015840101584010158401015840 + 212058311990810158401015840 = 0

(4)

together with

120579119860 = 0119908119860 = 01199081015840119860 = 0

Γ1205791015840119861 + 119896119890 (119908119861 + 119890120579119861) = 0minus119908101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) = 0

11990810158401015840119861 = 0

(5)


(120579 (119904 119905)119908 (119904 119905)) = (120579 (119904)

119908 (119904)) exp (120582119905) (6)


minus 1205822Λ2 120579 + Γ12057910158401015840 = 0 (7a)

1205822119908 + 1199081015840101584010158401015840 + 212058311990810158401015840 = 0 (7b)

120579119860 = 0 (7c)

119908119860 = 0 (7d)

1199081015840119860 = 0 (7e)

Γ1205791015840119861 + 119896119890 (119908119861 + 119890120579119861) = 0 (7f)

minus119908101584010158401015840119861 + 119896 (119908119861 + 119890120579119861) = 0 (7g)

11990810158401015840119861 = 0 (7h)





+ 1198623 (1120572 sin (120572119904) minus 1120573 sinh (120573119904)) (8)


1205722 = 120583 + radic1205832 minus 1205822


(9)


[B120579120579 B120579119908B119908120579 B119908119908

] c = 0 (10)





(11)



H119906 fl

119891119906 (120596 120583 119896 119890) = 0120597119891119906 (120596 120583 119896 119890)120597120596 = 0 (12)




minus( 1205822Λ2 + 119888119903120582)120579 + (Γ + 120577120582) 12057910158401015840 = 0(1205822 + 119888119905120582)119908 + (1 + 120578120582)1199081015840101584010158401015840 + 212058311990810158401015840 = 0

120579119860 = 0119908119860 = 01199081015840119860 = 0

(Γ + 120577120582) 1205791015840119861 + 119890 (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0minus (1 + 120578120582)119908101584010158401015840119861 + (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0

(1 + 120578120582)11990810158401015840119861 = 0

(13)



1205722 = 120583 + radic1205832 minus 120582 (119888119905 + 120582) (1 + 120578120582)1 + 120578120582



B119908120579 fl [119890 (119896 + 119888119887120582)120574 sin (120574) 0]119879 B119908119908



(14)


119891119889 (120582 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)fl 119891 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)

+ 119894119892 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0(15)



H119889 fl

119891(0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (16)




D fl

119891(0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (17)




4 Numerical Results





119888119903 = 1121198872119888119905120577 = 13120578

(19)


Λ = 10radic3Γ = 21 + ]

119888119903 = 119888119905300 120577 = 43120578

(20)







25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu









5 Conclusions






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)


010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002








References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of







+ 1198623 (1120572 sin (120572119904) minus 1120573 sinh (120573119904)) (8)


1205722 = 120583 + radic1205832 minus 1205822


(9)


[B120579120579 B120579119908B119908120579 B119908119908

] c = 0 (10)





(11)



H119906 fl

119891119906 (120596 120583 119896 119890) = 0120597119891119906 (120596 120583 119896 119890)120597120596 = 0 (12)




minus( 1205822Λ2 + 119888119903120582)120579 + (Γ + 120577120582) 12057910158401015840 = 0(1205822 + 119888119905120582)119908 + (1 + 120578120582)1199081015840101584010158401015840 + 212058311990810158401015840 = 0

120579119860 = 0119908119860 = 01199081015840119860 = 0

(Γ + 120577120582) 1205791015840119861 + 119890 (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0minus (1 + 120578120582)119908101584010158401015840119861 + (119896 + 119888119887120582) (119908119861 + 119890120579119861) = 0

(1 + 120578120582)11990810158401015840119861 = 0

(13)



1205722 = 120583 + radic1205832 minus 120582 (119888119905 + 120582) (1 + 120578120582)1 + 120578120582



B119908120579 fl [119890 (119896 + 119888119887120582)120574 sin (120574) 0]119879 B119908119908



(14)


119891119889 (120582 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)fl 119891 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)

+ 119894119892 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0(15)



H119889 fl

119891(0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (16)




D fl

119891(0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (17)




4 Numerical Results





119888119903 = 1121198872119888119905120577 = 13120578

(19)


Λ = 10radic3Γ = 21 + ]

119888119903 = 119888119905300 120577 = 43120578

(20)







25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu









5 Conclusions






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)


010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002








References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





1205722 = 120583 + radic1205832 minus 120582 (119888119905 + 120582) (1 + 120578120582)1 + 120578120582



B119908120579 fl [119890 (119896 + 119888119887120582)120574 sin (120574) 0]119879 B119908119908



(14)


119891119889 (120582 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)fl 119891 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578)

+ 119894119892 (120585 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0(15)



H119889 fl

119891(0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 120596 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (16)




D fl

119891(0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0119892 (0 0 120583 119896 119890 119888119903 119888119905 119888119887 120577 120578) = 0 (17)




4 Numerical Results





119888119903 = 1121198872119888119905120577 = 13120578

(19)


Λ = 10radic3Γ = 21 + ]

119888119903 = 119888119905300 120577 = 43120578

(20)







25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu









5 Conclusions






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)


010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002








References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of






119888119903 = 1121198872119888119905120577 = 13120578

(19)


Λ = 10radic3Γ = 21 + ]

119888119903 = 119888119905300 120577 = 43120578

(20)







25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu









5 Conclusions






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)


010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002








References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZu

ℋu

ℋu









5 Conclusions






25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)


010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002








References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋu

ℋd2

ℋd3

ℋd1

(a)

25

20

15

10

5

0

k

0 10 20 30 40 50 60 70

DZu

DZd2

DZd3

ℋd2

ℋd3

ℋd1

ℋu

(b)


010

008

006

004

002

000

ct

00 05 10 15 20 25 30 35

6 7 8 9

U S

= 1002








References























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





Journal of




flexural-torsional flutter and buckling of braced foil...

Documents