self-excited nonlinear response of a bridge...

BBAA VI International Colloquium on:Bluff Bodies Aerodynamics & Applications

Milano, Italy, July, 20–24 2008

SELF-EXCITED NONLINEAR RESPONSE OF A BRIDGE-TYPECROSS SECTION IN POST-CRITICAL STATE

Jirı Naprstek?, Stanislav Pospısil?, Rudiger Hoffer†, Joerg Sahlmen†

?Institute of Theoretical and Applied Mechanics, v.v.i.Academy of Sciences of the Czech Republic, Prosecka 76, 19700 Prague, Czech Republic

e-mails: [email protected], [email protected]

†Building Aerodynamics LaboratoryFaculty of Civil Engineering, Ruhr University Bochum Building IA 0/58, D-44780 Bochum, Germany

e-mails: [email protected], [email protected]

Keywords: aeroelastic system, self-excited vibration, limit cycles, instability, post-critical be-haviour

Abstract: Long slender civil-engineering structures exposed to wind, especially decks of sus-pension bridges, their supporting elements, masts, towers and tall buildings are susceptible tovibrations under certain circumstances due to pure wind load action or due to special aeroelas-tic effects. Interaction between structural response and wind load, having the influence on thestability, is nowadays of basic importance in advanced technical design of such structures.

Finding the stationary points and determining stability characteristics of a system is an im-portant undertaking in any attempt to understand or to modify its dynamical properties. Wedemonstrate the importance with an example describing the aeroelastic system with gyroscopic(non quadratic in the velocities) as well as with non-conservative terms. In the article we con-sider a non-linear system with two degrees of freedom representing an interaction of bendingand torsion of a slender beam vibrating in a cross flow. The shape makes possible to separateprincipal effects and the coupling of aeroelastic modes is caused solely by the flow around thestructure.

Conditions of existence of stationary points and their types are investigated using primar-ily the Lyapunov function and center manifold theory. The procedure presented is applicablefor Hamiltonian holonomic systems which are conservative, or nonconservative with certainlimitations on the generalized forces. The singular points are to be classified with respect totheir asymptotically stable or unstable character together with adequate physical interpretation.Attention is paid to attractive and repulsive areas surrounding these points. Using this back-ground several types of post-critical response in non-linear formulation will be presented, suchas stable/unstable limit cycles with various ratio of amplitudes of both components, or quasi-periodic response processes having a form of symmetric or asymmetric beating effects withstrong energy trans-flux between degrees of freedom.

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[email protected]

[email protected]

[email protected]

[email protected]

Jirı Naprstek, Stanislav Pospısil, Rudiger Hoffer, Joerg Sahlmen

1 INTRODUCTION

Slender line-like structures, such as long cable-supported bridges and light footbridges havetypically low eigen-frequencies and low damping, and are therefore sensitive to wind loading.Once such a structure starts to vibrate, a complex interaction between the moving boundaryand the airflow takes place, which may either effectively attenuate or reinforce the driving forceof the wind. This is the reason why the fundamental task of bridge design with regards to theaeroelasticity lies in the formulation of the self-excited forces and interaction with the response.From the theoretical point of view, this interaction leads to the origin of non-conservative forcescontributing to the stiffness matrix and in the same time to the origin of forces influencing adamping matrix in linear and non-linear way.

The design traditionally assisted by the studies in wind tunnels, is challenging in the sensethat there are many complicated issues to be considered. Experimental work in bridge aeroelas-ticity has been focused for long time on gaining the knowledge of so-called aeroelastic deriva-tives and on the critical state determination. This is nowadays the well known task, which how-ever does not comply with the real flutter nature, where the loss of stability is rather ”sudden”and various levels of the non-linearity strongly influence internal mechanism of self-excited os-cillations. On the linear (zero) level as a particular case, two parallel ways, each of them bring-ing some advantages, can be formulated. The duality of time and frequency domain formulationof the self-excited wind forces was and still is investigated, see e.g. [1]. In the time domain for-mulation of self-excited forces on a bridge deck, indicial functions are usually adopted, see e.g.[2], [3] and others. Aeroelastic problems including the modelling of aerodynamic forces excitedby non-stationary wind fields and considering non-linearities in both structural dynamics andaerodynamics are summarized in [4]. For the purpose of direct determination of critical veloc-ities in engineering computations the ”combined time/frequency” system can be used despitethe iterative methods are used. From mathematical point of view, however, such procedure isconsiderably problematic, as it implies too many assumptions which probably will not be com-plied with as soon as the response has lost the character of a purely harmonic movement withclearly expressed frequency [5].

The majority of the models have however either obvious or hidden linear character beingbased on various types of convolution formulations or other superposition related principles[6]. This is the reason why the analyses have revealed relatively considerable diversity of con-clusions in the basic terminology as well as in results based on experimental studies. Thereforeit is important, regarding the reliability of the system, to analyse also the post critical systembehavior, which cannot be represented using the linear approach only. Non-linear processesin the post critical regime are decisive from the point of view of a possible restabilization ofthe system due to non-linear forces, while linear configuration would lead to infinitely risingresponse due to positive real part of at least one eigen-value of the system matrix.

Moreover, the analysis has revealed relatively considerable diversity of conclusions basedon experimental studies as well as of the basic terminology. This is probably due to histor-ical treatment of these problems in a number of branches as well as due to the existence ofvarious types of instability domains and a number of bifurcation points. In the course of thisperiod, however, the research has succeeded in understanding most principal characteristics ofaeroelastic systems. Although some of these approaches are able to predict some lower limitsof aeroelastic stability loss, they avoid any possibility to investigate the post-critical behaviourwhich is of strongly non-linear character. However, the detailed knowledge of the post-criticalstate is very important being decisive from the viewpoint of a possible secondary restabilisationdue to non-linear effects. Consequently the conventional linear models are hardly acceptableand true non-linear approaches should be developed.

Another important factor consists in the possibility of origin of ”auto-parametric” resonance,

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particularly with the reference to the possible non-linear restabilisation. All these phenomenaare highly dangerous for the degradation of structure properties, as they result in a strong post-critical response containing superharmonic components, as a rule. In some cases such structuremay behave even as a selective resonator. On the other hand, some non-linear effects originatingfrom these interactions, however, can act like stabilizing factors. They are able at least for alimited period of time, to restitute one of lower types of stability after the structure has lost itsexponential or asymptotic stability, see [7], [8]. A number of partial phenomena resulting fromthe existence of parametric noises and asymmetry of the stiffness matrix (non-conservativeforces) and of the damping matrix (gyroscopic forces) have been described successfully in apurely theoretical way, [9], [10].

Besides the pure theory, there has been collected a large number of experimental worksrelated to this topic, see e.g. [11], [12]. Any of these branches creates its own experimentalconditions according to their needs with emphasis on their typical parameters. It is uncertainwhich parameters are fundamental for the particular instability origin and which are only typicalfor the particular technical branch. Due to the growth of the computer performance with anincrease in the computing power, modelling of the wind induced effects on a structure by usingthe principles of Computational Fluid Dynamics is much more frequent. However, it cannotreflect the inherent complexity of the fluid-structure interaction phenomena. Therefore, the roleof the experiments as the feedback source should not be suppressed. Moreover, the capacityof electronic equipment has an influence on the design of new experimental gadgets allowingvery precise measurement and data mining, which has not been possible in previous period. Asystematic research is needed to clarify the role of main individual parameters in the instabilityonset. It seems that in the given situation it is realistic to start with the development of auniform theory based simultaneously on experiments which would cover all known cases ofaeroelastic instability and possible restabilisation due to non-linear and non-symmetrical terms(vortex shedding incl. noises, galloping, flutter). Some of these phenomena may intermingleand generate further state not yet described theoretically.

2 MODEL OF THE STRUCTURE AND BASIC PROPERTIES OF THE SYSTEM

We start with reference to the vibration of a slender prismatic bar in the combination ofbending and torsion, according to the Fig. 1, circumvent by the wind along its longer side.

yϕ

(t)ϕ

xϕ

xS

S

xu(t)

y

V

Figure 1. Schematic depiction of the elastically supported girder with two degrees-of-freedom in the airstream.

Let us make use of the fact that this form is symmetrical along two axes, its center of torsionis identical with its centroid and the effects of aerodynamic forces can be referred to the centerof this section. The coupling of aeroelastic modes is caused solely by the flow around thestructure which produce the running lift L(t) and moment M(t) generated by heave and pitch

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oscillations.The problem to determine the aeroelastic forces (coefficients) has been dealt with in great

detail by some monographs (newly see e.g. [13]), in which the relation between the forcesand the air flow velocity has been described by linear equations. The aerodynamic coefficients,the root of the aeroelastic couplings, have to be a functions of reduced frequency κ = d ·ω/V to respect the fact emerging from the physical nature, that they are not static values andchanges with a frequency. For instance, the uplift force in this concept will have an entirelydifferent amplitude in case of harmonic movement with a low and a high frequency. From themathematical point of view, this approach can be justified until the response keeps the characterof purely harmonic motion.

3 BASIC EQUATION

Fluid-structure interaction of turbulent flow and oscillating sectional bridge models with twodegrees of freedom u and ϕ is often represented as a linear coupled system of common forcesof the structural part, buffeting forces FB due to wind gusts including resonance and aeroelasticforces Fae = [Lae,Mae]

T due to flow-structure interaction as shown in following equation:

FI + FC + FK = M · uT + C · uT + K · uT = FB(t) + [Lae(u, u, t),Mae(u, u, t)]T (1)

where u = (u, ϕ) is the vector of generalized displacements.In a mixed frequency and time-domain representation, the aeroelastic forces in vertical di-

rections Lae and the aeroelastic moment Mae due to aerodynamic stiffness and aerodynamicdamping contributions are commonly linked to linear terms of girder displacements and veloc-ities via flutter derivatives. Such description, however, is linear and can be used to investigatethe transition and especially post-critical state only with certain limitations. It is possible toconsider the lifting force and the moment as a certain functions of generalized displacement u,rotation ϕ, and their derivatives. For the linear parts expressions, as for example in [14] canbe used. If the nonlinear part has to be included, it can be assumed in a form of a higher de-gree polynomial which represents a natural extension of the linear approach. This formulationresults from the experience with experimental investigations in a wind channel performed inpost-critical regimes. Omitting the buffeting forces FB(t) and using some arrangements weformulate the equations of motion as proposed in [15]:

u+ 2ωbuu+ ω2uu = Km(1− γuuu

2 − γuϕϕ2)(buuu+ buϕϕ) +

+Km(1− βuuu2 − βuϕϕ

2)(cuuu+ cuϕϕ) (a)

ϕ+ 2ωbϕϕ+ ω2ϕϕ = KJ(1− γϕuu

2 − γϕϕϕ2)(−bϕuu+ bϕϕϕ) +

+KJ(1− βϕuu2 − βϕϕϕ

2)(−cϕuu+ cϕϕϕ) (b)

(2)

or alternatively:

u+ 2ωbuu+ ω2uu = Km(1− βuuu

2 − βuϕϕ2)(cuuu+ cuϕϕ+ buuu+ buϕϕ) (a)

ϕ+ 2ωbϕϕ+ ω2ϕϕ = KJ(1− βϕuu

2 − βϕϕϕ2)(−cϕuu+ cϕϕϕ− bϕuu+ bϕϕϕ) (b)

(3)

in which ωu is the eigenfrequency of the vertical motion, ωϕ is the rotational eigen-frequencyand ωbu, ωbϕ are the damping ratios. The coefficients βij and γij (i, j = u, ϕ) represent scalingfactors of the non-linear part and form square asymmetric matrices. The parameters cij, bij formsquare asymmetric matrices as well and could be considered as generalizations of very well-known aeroelastic derivatives and should be identified experimentally. Coefficients KJ , Km

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depend on the cross-section geometry and the characteristics of the wind:

Km =1

2m%V 2 · 2d ; KJ =

1

2J%V 2 · 2d2 (4)

where V is the wind velocity, d is a characteristic dimension of the girder cross section and % isthe air density under typical condition.

Introducing external excitation to the right hand side of the Eq. (2a), the component differentfrom zero will be the displacement u(t) while the rotation ϕ(t) may identically equal zerounder certain conditions. Similar result will be attained introducing external excitation onlyto the right hand side of the Eq. (2b) and analogously for Eqs. (3). This will create so-calledthe reduced problem of aeroelastic stability, useful also in practical applications in studyingself-induced vibration of structures with one component of motion.

Roughly speaking the models (2) and (3) are oriented on the non-linearities which are de-scribed by velocities and displacements. In this meaning they can be compared to the equationof motion of a single-degree-of freedom system with non-linear stiffness part (Duffing), com-bined with the equation where non-linear damping depends on the square of motion velocity(Rayleigh) or, in contrast, combined with the equation where the effective damping dependson the square of displacement (Van der Pol). Consequently, we name the system (2) Rayleigh-Duffing and system (3) Van der Pol-Duffing.

Besides general stability analysis (based on local linearizing methods) we are looking forthe birth or a death of a periodic orbit through a change in the stability of an equilibrium. Anequilibrium point is said to be hyperbolic, if all the eigenvalues of the Jacobian matrix havenonzero real parts. On the other hand when the eigenvalues of the linearized vector field atan equilibrium point are purely imaginary, the local dynamics about the equilibrium cannot bedetermined by the linear approximation and higher order terms should be analyzed.

4 INSPECTION OF STABILITY

Conclusions about the stability of a critical point may be acquired by means of constructionof a suitable auxiliary function, called Lyapunov function Φ. Its total time derivative Ψ, can beidentified as the rate of change of Φ along the trajectory of the system that passes through thepoint of interest in the phase plane. There are no general methods of construction of Lyapunovfunction and often the judicious trial-and-error approach may be necessary. However, the useof some general properties of mechanical systems often gives good results. For instance, toolsbased on energy balance and the first integrals are usually very effective. Some ideas leading tothis procedure can be found e.g. in the monograph [16]. If a system has several first integrals, theLyapunov function can be written in a form of a linear combination of first integrals and possiblyof their functions. The coefficients of these linear combinations, which can be considered asLagrange multipliers, must be so determined as to get to the resulting function the properties ofpositive definiteness.

If we move an equilibrium point to the origin of the state space by linear transformations,then the following stability criteria apply:

i If Φ(x) is positive for all values of the state variables within a region except at the origin,where Φ = 0, then this scalar function is called positive definite in that region.

ii If the time derivative of Φ(x) is negative for all values of the state variables within aregion except the origin, then the scalar function is called negative definite in that region.

then the system is considered to be asymptotically stable within that region. If conditions (i)and (ii) are true in the entire region of the state space, then the system is globally asymptotically

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stable. Simply, this criteria is a restatement of the energy requirements of stability for simplemechanical systems. Criteria (ii) indicates that as time increases the system must not increaseits energy, because it could then drift out of the stable region. Finding a suitable Φ function wasfor the most part a matter of guessing until the development of the systematic method of usingthe gradient of Φ to determine a suitable Lyapunov function appeared, see e.g. [17].

It is assumed that there exists potential vector field with the gradient ∇Φ and it may bewritten:

Φ(x) =dΦ

dt= (gradΦ)T x (5)

Lyapunov function can be found from the gradient of Φ by the integration across the statespace. This integration will be independent of the path if rotΦ = 0 which is equivalent to the

∂(∇Φ)i

∂xj

=∂(∇Φ)j

∂xi

(6)

for i = 1, 2 · · ·n. The gradient of Φ is searched in the form ∇Φ = αx where α is an n × nmatrix of unknown coefficients. Finally, to generate the Lyapunov function, using this methodwe form dΦ/dt as an equation:

Φ = αxAx (7)

and utilizing the constraint that dΦ/dt must be negative definite, we obtain values for the αi,j .This method will be used hereafter.

In the cases, where the first integral of the system cannot be found, the stability of the pointin the local sense can be sometimes established using the center manifold theorem, see e.g.[16], which is used to reduce the dimension of the space variables and the stability of the non-linear system can be analysed at the center manifold. It is invariant and can be expressed (intwo-dimensional space) for example by a power series.

4.1 Rayleigh-Duffing versus Van der Pol-Duffing equation

Let us concentrate on the case when the motion is allowed in rotation only. Though sim-plified, this assumption allows us to analyze some important properties and to understand therole of individual parameters in the system. Investigation of the stability of nonlinear systemaround stationary points can be started by means of the corresponding linear system. However,for example, no conclusion can be drawn when the critical point is a center of the correspondinglinear system. Also, for an asymptotically stable critical point it may be important to investigatethe domain of asymptotic stability.

Respecting assumption mentioned above the Rayleigh-Duffing system can be written in thefollowing reduced form:

ϕ = ψ

ψ = − [2ωbϕ −KJbϕϕ (1− γϕϕψ2)]ψ −

[ω2

ϕ −KJcϕϕ (1− βϕϕϕ2)]ϕ

(8)

whereas Van der Pol-Duffing version of the system would read:

ϕ = ψ

ψ = − [2ωbϕ −KJbϕϕ(1− βϕϕϕ2)]ψ −

[ω2

ϕ −KJcϕϕ(1− βϕϕϕ2)]ϕ

(9)

The position of the critical points can be obtained solving the respective algebraic systemswhich follows from Eq. (8) or Eq. (9) by demanding, that the first derivatives on the left-handside of equations vanish. Each of these systems admits three independent solutions. The first

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one corresponds to the case, when both the velocity and the rotation vanish providing the trivialsolution:

P1 : ϕ = 0 ; ϕ = ψ = 0 (10)

The second and the third solution implies that the velocity vanish as it follows from the firstequation, while the second equation leads to the formula:

P2,3 : ϕ = ±√

(KJcϕϕ − ω2ϕ)/(KJβϕϕcϕϕ) = ±a ; ϕ = ψ = 0 (11)

So finally, there are three equilibrium points P1 = (0, 0) and P2,3 = (±a, 0). The points P2 andP3 are symmetrical with respect to the origin, so that it is sufficient to analyze the stability ofone of them only. In order to analyze the equilibrium stability at the point P2, the transformationof the coordinates is useful. We use the transformation which shifts the origin to the point a asit is depicted on the Fig. 2. It is given by the formulas:

ϕ = a+ ξ ; ψ = 0 + ζ (12)

−2a −aa−a

ϕ

ψ

ξ

ζ

Figure 2. The transformation of coordinates used in the case of Eq. (8).

For the system of transformed Rayleigh-Duffing type of differential equation (8) then holds:

ξ = ζ

ζ = − [2ωbϕ −KJbϕϕ(1− γϕϕζ2)] ζ −

[ω2

ϕ −KJcϕϕ(1− βϕϕ(ξ + a)2)]

(ξ + a)(13)

4.2 Stability analysis

One may see, that there are three equilibrium points P1, P2 and P3. The linear variationalequation of the vector field given by the right side of the Eqs (8), or (9) leads to the characteristicequation and to the eigenvalues of the Jacobi matrix:

J =

[0 ; 1−[ω2

ϕ − cϕϕKJ(1− 3βϕϕϕ2)]

; − [2ωbϕ − bϕϕKJ(1− 3γϕϕψ2)]

](14)

Characteristic polynomial for the equilibrium point P1 is given by following formula:

∆ = λ2 + λ(2ωbϕ − bϕϕKJ) + ω2ϕ − cϕϕKJ (15)

Similarly, at the point P2 the characteristic polynomial is given by:

∆ = λ2 + λ(2ωbϕ − bϕϕKJ)− 2ω2ϕ + 2cϕϕKJ (16)

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Eigenvalues in the equilibrium points can be now easily calculated. They are necessary in theevaluation of the stability of the non-linear original system except some (critical) cases. In P1

they can be calculated from the Eq. (15) and are given by the following formula:

λ1,2 = −1

2(2ωbϕ − bϕϕKJ)± 1

2

√(2ωbϕ − bϕϕKJ)2 − 4(ω2

ϕ − cϕϕKJ) (17)

For the eigenvalues in the point P2 a similar formula can be derived using Eq. (16):

λ1,2 = −1

2(2ωbϕ − bϕϕKJ)± 1

2

√(2ωbϕ − bϕϕKJ)2 − 8(ω2

ϕ − cϕϕKJ) (18)

The stability analysis is based on the Jacobi matrix (14) and the Lyapunov function. We expressand plot some particular examples to see, how the course of the Lyapunov function derivativechange on the trajectory, representing the solution of the system and therefore also to acknowl-edge, whether the stationary points are stable or not. Certainly, this phenomenon depends onthe selection of parameters. As it follows from Figs. 3–9, the Lyapunov functions Φ and theirderivatives Ψ are represented by the contour lines. The dashed lines stand for negative val-ues, on the bold lines the functions vanish and on the continuous lines the function values arepositive.

We observe that two parameters are important in order to judge the stability of the stationarypoints. In the case of Rayleigh-Duffing Eq. (8) they are given by the formulas:

r1R = 2ωbϕ −KJbϕϕ and r2R = ω2ϕ − cϕϕKJ . (19)

In case of Van der Pol-Duffing Eq. (9), we obtain analogous (though not exactly the same)parameters, which we denote r1V and r2V .

The Fig. 3 demonstrates the example of Lyapunov function and its time derivative for thesystem described by the Eq. (8) in the vicinity of the point P1. The equations for them are givenas follows:

Φ =1

3(2ϕ2 + 5ϕψ+ 2ψ2) ; Ψ =

1

3(−5ϕ4− 4ψϕ3 + 5ϕ2 + (8ψ− 5ψ3)ϕ− 4ψ4 + 5ψ2) (20)

The function Φ is positive everywhere around the origin, while the function Ψ is negative.On the Fig. 5 the example of functions in the vicinity of the point P2 is demonstrated. In this

case, the equilibrium point is the origin of the new coordinate system ξ, ζ . The formulas for thefunction are:

Φ =1

4(5ξ2+2ξζ+3ζ2) ; Ψ =

1

2(−ξ4−3ξ3(1+ζ)−ξ2(2+9ζ)−ξζ(2+ζ2)−ζ2(2+3ζ2)) (21)

Two other examples are demonstrated at the Fig. 7 and Fig. 9, where the Lyapunov functionsand their derivatives relevant to the Van der Pol-Duffing system are shown. For the vicinity ofthe point P1 they are given by formulae:

Φ = 2ϕ2 − ϕψ + ψ2 ; Ψ = ϕ4 + ϕψ − ϕ3ψ + ψ2 + ϕ2(1− 2ψ2) (22)

while in the local neighborhood of the point P2 these functions are as follows:

Φ =1

4(9ξ2−2ξζ+3ζ2) ; Ψ =

1

2(ξ4 +ξ3(3−2ζ)+2ξ(1−3ζ)ζ+2ζ2 +ξ2(2−7ζ−3ζ2)) (23)

From this formulas and figures below, we observe that for particular selection of parametersr1 and r2 from the Eq. (19) the function Φ have different course. It may represent the convex

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surfaces or it can represent the saddle where the stationary point is not stable anymore. This isdemonstrated at the selected figures below for particular cases of Eqs. (8) and (9).

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

2

3Ζ

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

2

3Ζ

Figure 3. An example of the Lyapunov function (left) and its derivative (right) for the system (8) in theneighborhood of the point P1. r1R = 0, r2R > 0 . The equilibrium point is unstable.

-2 -1 1 2Ξ

-2

-1

1

2Ζ

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

2

3Ζ

Figure 4. An example of the Lyapunov function (left) and its derivative (right) for the system (8) in theneighborhood of the point P1. r1R > 0, r2R > 0. The equilibrium point is stable.

-1.5 -1.0 -0.5 0.5 1.0 1.5Ξ

-2

-1

1

2

Ζ

-2 -1 1 2 3Ξ

-3

-2

-1

1

Ζ

Figure 5. An example of the Lyapunov function (left) and its derivative (right) for the system (8) in theneighborhood of the point P2. r1R > 0, r2R < 0. The equilibrium point is stable.

-1.0 -0.5 0.5 1.0Ξ

-2

-1

1

2

Ζ

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

Ζ

Figure 6. An example of the Lyapunov function (left) and its derivative (right) for the system (8) in theneighborhood of the point P2. r1R < 0, r2R < 0. The equilibrium point is unstable.

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-1.0 -0.5 0.5 1.0Ξ

-1.5

-1.0

-0.5

0.5

1.0

1.5

Ζ

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

2

3Ζ

Figure 7. An example of the Lyapunov function (left) and its derivative (right) for the system (9) in theneighborhood of the point P1. r1V < 0, r2V > 0. The equilibrium point is unstable.

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

2

3Ζ

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

2

3Ζ

Figure 8. An example of the Lyapunov function (left) and its derivative (right) for the system (9) in theneighborhood of the point P1. r1V > 0, r2V > 0. The equilibrium point is unstable.

-1.0 -0.5 0.5 1.0Ξ

-2

-1

1

2

Ζ

-3 -2 -1 1 2 3Ξ

-3

-2

-1

1

2

3Ζ

Figure 9. An example of the Lyapunov function (left) and its derivative (right) for the system (9) in theneighborhood of the point P2. r1V > 0, r2V > 0. The equilibrium point is unstable.

4.3 Pure imaginary eigenvalues and bifurcation

Let us investigate the stability and bifurcation of a stationary points of the system (8) in casesof varying parameters r1R, r2R. A special case is when the eigenvalues of the relevant Jacobihave zero real parts which occurs, see Eq. (17), if one of the conditions (19) (or both) equalzero. Moreover, if the eigenvalues of the Jacobi matrix cross transversally the imaginary axis,the change of stability (bifurcation) character occurs. This phenomenon is demonstrated below.There exist ”degenerated” cases of governing equations. One example is given here:

ϕ = ψ

ψ = −bϕϕKJγϕϕψ3 − (ω2

ϕ −KJcϕϕ)ϕ− βϕϕKJcϕϕϕ3

(24)

which results from Eq. (8), when the coefficient r1 equals zero.

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The center manifold theorem may be used when the Jacobi matrix has one eigenvalue withzero real part and one with negative real part. It represents the existence of non-hyperbolicequilibria of the related non-linear system. For example, let us analyze the Eq. (8), assumingr2R = 0 and r1R > 0. The equation can be transformed into the system with the linearizationabout the origin being in Jordan normal form with new variables u(t), v(t). It is obtained bythe similarity transformation using eigenvectors of the Jacobi matrix. The stability can be thenanalysed on a invariant curve v = g(u) for which v = g′u - local center manifold - expandedinto a approximative power series in the variable u, where the coefficients of the series are to bedetermined.

The transformed Eq. (8) is written:

u = −(u3 + 3u2v + 3uv2)

v = −v + (u3 + 3u2v + 3uv2)(25)

Introducing the proposed series into the Eq. (25) and performing some mathematical arrange-ments, we obtain the equation for dynamics on the center manifold given by the more simplerfirst order differential equation to be analyzed:

u = −u3 − 3u5 +O(u6) (26)

Doing this, we observe, that the origin is asymptotically stable with the slow non-exponentialapproach to the equilibrium, because of the non-hyperbolicity. The graph on the left on theFig. 10 shows the phase diagram as a solution of this case.

-0.5 0.5 1.0j

-1.0

-0.5

0.5

1.0Ψ

-0.2 0.2 0.4 0.6 0.8 1.0j

-0.5

0.5

1.0Ψ

-1.0 -0.5 0.5 1.0j

-1.0

-0.5

0.5

1.0Ψ

Figure 10. Phase diagrams for ”degenerated” equation of motion, given by the Eq. (24).

Another special example arises, when both r1 = 0 and r2 = 0. The origin is still asymptot-ically stable and the approach to the equilibrium is again non-exponential due to the structureof the Eq. (24), see middle graph of the Fig. 10. Mechanically it corresponds to the weak (zerolinear stiffness) spring of the Duffing oscillator near the equilibrium. The right graph of thesame figure represents the event, when when r1 = 0 and r2 < 0. Eigenvalues are real withidentical absolute value. The origin P1 is unstable and there are two other asymptotically stableequilibrium points P2,P3.

On the Fig. 11 the supercritical limit cycle birth is shown. Passing with eigenvalues throughthe imaginary axis transversally, we get instable origin and a limit cycle respecting the condi-tions r1 < 0 and r2 > 0. The right diagram demonstrates the behaviour in the case of conditionsr1 < 0 and r2 < 0.

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-1.0 -0.5 0.5 1.0j

-1.0

-0.5

0.5

1.0

Ψ

-2 -1 1 2j

-1.0

-0.5

0.5

1.0

Ψ

Figure 11. The oscillation with occurrence of the supercritical limit cycle.

5 EXPERIMENTAL SOLUTION

The experimental and numerical prediction of flutter instability is of important concern forthe design of flexible structures. Experiments focused upon the determination of the criticalstate is therefore a common task, which however does not reflect the instability nature, which is”sudden” and may undergo destructive motion with large amplitudes and velocities, or on theother hand, it may sometimes have self-stabilizing character due to the non-linear effects. Fromthe point of view of basic research as well as from the perspective of the structural serviceabilityand lifetime, it is interesting therefore to get information predominantly regarding the systembehaviour not only before the crisis appears, but also during the transition time as well as in thepost-critical state.

55 60 65 70t

-0.02

-0.01

0.00

0.01

0.02

0.03

u

55 60 65 70t

-0.2

-0.1

0.0

0.1

0.2

j

Figure 12. Response in vertical motion and rotation obtained by experiments. Due to the influence ofnon-linear terms, the oscillation has several bumps before the limit cycle oscillation starts.

Several experiments have been done in order to investigate the system behaviour for varyingequation parameters and initial conditions. The time history of the response is very sensitive toeven slight change of them. An example of experimental results obtained from the measurementin the wind tunnel is given on the Fig. 12. The tested girder with rectangular cross section hasbeen investigated. Around the equilibrium points, the response amplitudes rise dramatically,stabilize and rise again and energy transmission between two modes is identified.

Using experimental rig described in [18], the forces and amplitudes on several section mod-els have been measured not only during the low amplitude vibration, but also during and afterthe instability onset. The wind speed has been increasing continuously during the wind tun-nel experiments. Pitch motion of the deck was small until it reached the bifurcation point.Keeping the wind speed constant, it reached the limit cycle apparently of Rayleigh-Duffingtype. The amplitudes of both the heave and pitch motions were registered using accelerome-ters. This strategy enabled to analyze not only the response of the deck but also the pressures

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in the individual points on the surface. It has been related to the large amplitude response ofthe bridge-like structure with rectangular cross-section and aspect ratio 1:5 using a newly de-veloped testing rig (Fig. 13). By the system of mechanically independent adjustable torsionalsprings, the frequencies of the girder can be tuned to theoretically arbitrary values. Pressuresfrom twelve sensors located on a surface of the deck was recorded during experiments for theidentification purposes.

Figure 13. New test rig used by authors for identification of non-linear aeroelastic coefficients.

Figure 14 represents the time shot, when the excessive amplitude growth started during thetransition through the critical (bifurcation) state. Especially rotation, a driving DOF, exceededthe angle, where the linear assumptions are unsatisfactory. The red (dash-dotted) lines are thepressures on the top surface of the beam. The blue (dashed) line represents the pressures on thebottom face of the body. The positive pressures are pointing out of the body.

9 10 11 12

2 3 4

5

1

6 7 8

Figure 14. Large amplitude response of the system in wind tunnel. The pressures on the surface of thebeam are indicated. Pressures pointing out of the body are positive.

Excessive responses occurred when the flow speed in the wind tunnel was near to, or iden-tical to the critical velocity. Pure heave was damped out, and periodic limit cycle oscillationsoccurred in the coupled mode shape consisting of both heave and torsional component. The va-lidity of the Eq. (1) is limited to this range. The amplitudes grow drastically for Vred > Vred,crit,so that the squares of the amplitudes and the coefficients βij and γi,j (i, j = u, ϕ) can gainimportance.

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5 10 15 20t

-1.0

-0.5

0.0

0.5

1.0

j

5 10 15 20t

-0.2

-0.1

0.0

0.1

0.2

u

Figure 15. Linear identification of aeroelastic derivatives at the rectangular girder for the displacement(left) and rotation (right).

The pressures measured at the model surface has been used for the identification of flutterderivatives according to the procedure in [19]: the responses measured in the wind tunnel hasbeen introduced into left hand sides of the equation (2). In the next step, the right hand sidesrepresenting the aeroelastic pressures have been established. The comparison of left hand sides(blue line) with the right hand sides (red line) is available from the Fig. 15. It can be stated, thatin the case of vertical movement u, the Eq. (2a), both curves are in good accord - left graph.Comparing the curves of the rotation ϕ, equation (2b) - right graph, the differences between bothsides are apparent. This is caused by the usage of linear identification method. Its insufficiencyunder common condition in emphasized more significantly in the case of rotational motion ϕthan in heave u.

6 CONCLUSIONS

The non-linear mathematical model of the bending-torsional flutter has been composed andverified considerably. It includes the linear approach being realistic for low air-stream veloci-ties when the system finds in a sub-critical stable regime. Duffing, Rayleigh and van Der Polnon-linear terms in multi-component form with cyclic symmetry have been introduced into thedifferential system. They act as consistent expansion of the linear approach. The process of themathematical model composition followed rather intuitive steps and trials to present an inher-ent description of effects and processes known from experimental measurements and numerical2D/3D simulations.

The theoretical model of the bending-torsional flutter shows the considerable sensitivity ofthe system self-excited vibration. After the loss of stability of trivial solution the responsein one degree-of freedom tends to stabilize itself the form of approximate harmonic solution.The growth of this component after the loss of stability of trivial solution is prevented by asignificant energy barrier. The system tends to escape to semi-trivial response into the othercomponent. When this energy barrier has been overcome, however, the system loses stabilityand its response grows beyond all limits. This typical case has a number of intermediate stepsand special states, the origin of which depends on the appropriate combination of parametersdescribed in the text.

The non-linear mathematical model of the bending-torsional flutter has been composed andverified considerably. It includes the linear approach being realistic for low air-stream veloci-ties when the system finds in a sub-critical stable regime. Duffing, Rayleigh and van Der Polnon-linear terms in multi-component form with cyclic symmetry have been introduced into thedifferential system. They act as consistent expansion of the linear approach. The process of themathematical model composition followed rather intuitive steps and trials to present an inher-

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ent description of effects and processes known from experimental measurements and numerical2D/3D simulations.

The basic properties of the non-linear system have been investigated when passing into thenon-stable domain. Possibilities of the post-critical re-stabilisation on the level of stable limitcycles (if any) are discussed. The widely known effects typical for post-critical regimes ofvarious types can be described by means of the proposed non-linear model. In particular pitch-fork bifurcation points have been detected with obvious configuration of stable and unstablebranches; conditions of existence and relevant portraits of the principal limit cycles have beencarried out; energy transflux between individual degrees of freedom has been detected andquantified. In order to enable the analysis of the system behaviour in various post-critical con-ditions, stationary points have been located and their near neighbourhood carefully investigatedby means of Jacobi matrices to identify their character and consequently the correspondingreaction of the system.

After the loss of stability of trivial solution the response in one degree-of freedom tends tostabilize itself in the form of approximate harmonic or polyharmonic solution. The growth ofthis component after the stability loss of the trivial solution is prevented by a significant energybarrier. The system tends to the response interacting with the other component. When thisenergy barrier has been overcome, however, the system loses stability and its response growsbeyond all limits. This typical case has a number of intermediate steps and special states, theorigin of which depends on the appropriate combination of parameters described in the text.

Lyapunov method together with the center manifold theory have approved their capabilitiesto provide an overview concerning behaviour of the system under large variety of geometricand physical input parameters and initial conditions. The results obtained in the paper dependto a considerable extent on the parameters introduced. They should be identified from the mea-surements by means of new non-conventional methods combining numerical and experimentalprocedures. The development of these methods is in progress and relevant results will be pro-vided in the near future. The more advanced ”non-linear” identification procedures must beemployed in order to identify the aeroelastic coefficients introduced in the proposed non-linearmodel. The realized experiments on the newly developed experimental facilities and test rigswill be focusing on that.

ACKNOWLEDGMENT

The support of Grants No. 103/07/J060, 103/06/0099, A207 1401, Grant DFG 476 TSE113/52/0-1 and the research plan AV 07020710524 (ITAM) is acknowledged.

REFERENCES

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[3] L. Caracoglia, N. P. Jones. A methodology for experimental extraction of indicial func-tions for streamlined and bluff deck sections. Journal of Wind Engineering and IndustrialAerodynamics, 91: 1135–1150, 2003.

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[4] X. Chen, A. Kareem. New frontiers in aerodynamic tailoring of long span bridges: anadvanced analysis framework. Journal of Wind Engineering and Industrial Aerodynamics,91, 1511–1528, 2003.

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[19] F. Ricciardelli, E.T. de Grenet. Evaluation of bridge flutter derivatives from wind ex-cited vibration section models. Structural Dynamics EURODYN 2002 (H. Grundmann,G.I. Schueller eds.), Munich, 589–592, 2002.

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