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1 | L e T h a i H o a – F l u t t e r s t a b i l i t y a n a l y s i s : T h e o r y & E x a m p l e

FLUTTER STABILITY ANALYSIS

THEORY AND EXAMPLE

Prepared by Le Thai Hoa

2004

2 | L e T h a i H o a – F l u t t e r s t a b i l i t y a n a l y s i s : T h e o r y & E x a m p l e

FLUTTER STABILITY ANALYSIS

1. INTRODUCTION

There are two typical types of bridge flutter were i) Torsional flutter that the

fundamental torsional mode dominantly involves to the flutter instability ii) Coupled flutter

that the fundamental torsional mode aerodynamically couples tendency with either of any first

symmetric or ansymmetric heaving mode at single frequency (called flutter frequency) and

also known as the so-called classical flutter (similarly to flutter of airfoil wings). Various

experiments and numerical analyses [Matsumoto et al.(1996,1997)] showed that,

moreover, the torsional flutter seems to dominate almost cases of bridges with bluff

bridge sections as low slenderness ratio (B/D) rectangular sections, H-shape sections,

stiffened truss sections, whereas streamlined boxed bridge sections are favorable for

coupled flutter. However, the Akashi-Kankyo bridge exhibited with coupled flutter

that this is never experienced before with stiffened truss sections.

Flutter generation mechanism might be more difficult, however, by uses of series of

experiments on various fundamental sections and based on flow-structure interaction

phenomena as local separation bubble, reattachment, vortex shedding on structural

surface that Matsumoto et al. (1996,2000) classified the mechanism of flutter

instability generation of 2D H-shaped and rectangular sections into detailed

branches: i) Low-speed torsional flutter, ii) High-speed torsional flutter, iii) Heaving-branch

coupled flutter, iv) Torsional-branch coupled flutter and coupled flutter, v) Heaving-torsional

coupled flutter.

Flutter problems can be approximately divided by analytical and experimental

methods and simulation. Experimental approach is thanks to free vibration tests on

3 | L e T h a i H o a – F l u t t e r s t a b i l i t y a n a l y s i s : T h e o r y & E x a m p l e

2D bridge sectional model in wind tunnel laboratory. Computational fluid dynamics

(CFD) technique has gained much development so far to become useful supplemental

tools beside analytical and experimental methods and it is also predicted broadly that

such the CFD might replace wind tunnel tests in future, however, this simulation

method still has many limitations to cope with complexity of bridge sections and

nature of 3D bridge structures.

Fig. 1 Branches for flutter instability problems

Bleigh(1951) introduced empirical formula to calculate critical flutter velocity of

2DOF flutter problem for airfoil and thin-plate sections, Selberg(1961) developed

Bleich’s formula by putting the shape ratio to apply for various types of bridge

sections, moreover, Kloppel(1967) exhibited under a form of empirical diagrams.

Theodorsen(1935) applied potential theory of airfoil aerodynamics by introducing so-

called Theodorsen’s circulation functions to model self-controlled flutter forces,

meanwhile Scanlan(1971) used experimental approach to build such the self-

controlled forces by so-called flutter derivatives. Because the potential theory

validates in certain conditions of non-separation and non-reattachment around

Analytical Methods Empirical Formula 2DOF FlutterSolutions

nDOF FlutterSolutions

Selberg’s; Kloppel’s ComplexEigenMethod Step-by-Step Method

Simulation Method

Single-Mode Method

Multi-mode Method

Computational Fluid Dynamics (CFD)

Free Vibration Method

Flutter problems

Experiment Method

Two-Mode Method

4 | L e T h a i H o a – F l u t t e r s t a b i l i t y a n a l y s i s : T h e o r y & E x a m p l e

structural sections, thus the Theodorsen’s self-controlled flutter forces are limitedly

applied only on flutter problems of airfoil and thin-plate structures, thus Scanlan’s

ones are widely applied so far for flutter analytical problems of 2DOF systems and 3D

bridge structures with various types of cross-sections.

For 2DOF flutter problems, there are two powerful analytical methods: the complex

eigenvalue method [Simui&Scanlan(1976)] and the step-by-step method

[Matsumoto(1995)]. Though the complex eigenvalue method has been applied for a

long in 2DOD flutter problems, but difficulty to investigate relationship of system

damping ratio, system frequency on wind velocity, inter-relation between flutter

derivatives as well, the step-by-step method is very favorable over such the above-

mentioned limitations to clarify a role of flutter derivatives on critical condition and

on flutter stabilization.

For analytical methods for bridge or nDOF systems’ flutter problems, there are two

approaches: i) finite differential method (FDM) in linear-time approximation and ii) finite

element method (FEM) in modal space. However, the most state-of-the-art development

of analytical methods has carried out in the later. Agar(1989) developed FDM for

flutter problem of suspension bridges. Scanlan(1987,1990) firstly introduced sing-

mode and two-mode flutter analytical methods thanks to generalized transforms and

modal technique and based on idea that critical flutter conditions are prone to

dominant contribution of fundamental torsional mode (torsional flutter) or of

coupling between two torsional and heaving modes (coupled flutter). Many recent

studies [Pleif et al(1995), Katsuchi(1999), Ge et al.(2002)], however, pointed out that

in many cases of bridges there are not the fundamental torsional and heaving modes

involved to the critical flutter conditions, but many modes (multi-mode method)

superpose to generate more critical conditions.

5 | L e T h a i H o a – F l u t t e r s t a b i l i t y a n a l y s i s : T h e o r y & E x a m p l e

2. SINGLE TORSIONAL FLUTTER PROBLEM

The 1DOF motion equation of the torsional flutter (: torsional motion) can be written

as follow:

][ 2 1 *

3 2*

2 22 AKU

BKABUKCI

(A1.1)

Transforming above equation to the ordinary form as:

* 3

2* 2

22*

2 12 AK

U BKABU

I

(A1.2)

Where: 2 = I K ;

IK C

.2

(A1.3)

We have the equation:

0) 2 1()

2 12( *3

2222* 2

22 AKBUIU BKABU

I

0) 2 1()

4 1(2 *3

2222* 2

3

AKBUI KABU

I (A1.4)

For simplifying, we can write:

02 2 (A1.5)

We easily write the solution of above equation under following form:

)sin( 0 tAe t

Thus, the instability condition of the single torsional flutter follows:

0 or 0)4 1( *2

3 KABU I

(A1.6)

* 2

3

4 1 KABU

I

As a result, KUB

IA 3 * 2

4

(A1.7)

Through above unequality, the significant role of the torsional-motion-related flutter

derivative *2A (aerodynamic damping force) can be clearly approved.

6 | L e T h a i H o a – F l u t t e r s t a b i l i t y a n a l y s i s : T h e o r y & E x a m p l e

3. COMPLEX EIGENVALUE FLUTTER PROBLEM FOR 2DOF HEAVING-

TORSIONAL MOTION EQUATION SYSTEM

The 2DOF heaving and torsional motion equations of the flutter (h: heaving motion,

: torsional motion) can be expressed as follow:

][ 2 1 *

3 2*

2 * 1

2

HK U BKH

U hKHBUhKhChm hh

(A2.1)

][ 2 1 *

3 2*

2 * 1

22

AKU BKA

U hKABUKCI

(A2.2)

Transforming above equations to the ordinary form:

*3

2* 2

* 1

22

2 12 HK

U BKH

U hKHBU

m hhh hhh

(A2.3)

* 3

2* 2

* 1

22*

2 12 AK

U BKAhKABU

I

(A2.4)

Where: 2h = m Kh ; 2 = I

K ; mK