Flutter stability thle/Flutter stability FLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE

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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