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Page 1: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

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

Page 2: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

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

Page 3: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

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

Page 4: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

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.

Page 5: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

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:

][21 *

32*

222 AK

UBKABUKCI

(A1.1)

Transforming above equation to the ordinary form as:

*3

2*2

22*

212 AK

UBKABU

I

(A1.2)

Where: 2 =

IK ;

IKC

.2

(A1.3)

We have the equation:

0)21()

212( *

32222*

222 AKBU

IUBKABU

I

0)21()

41(2 *

32222*

23

AKBU

IKABU

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 KABUI

(A1.6)

*2

3

41 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.

Page 6: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

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:

][21 *

32*

2*1

2

HKUBKH

UhKHBUhKhChm hh

(A2.1)

][21 *

32*

2*1

22

AKUBKA

UhKABUKCI

(A2.2)

Transforming above equations to the ordinary form:

*

32*

2*1

22

212 HK

UBKH

UhKHBU

mhhh hhh

(A2.3)

*3

2*2

*1

22*

212 AK

UBKAhKABU

I

(A2.4)

Where: 2h =

mKh ; 2

= I

K ; mK

C

h

hh .2 ;

IKC

.2

(A2.5)

Introducing time-dimensionless variable: s = B

Ut (A2.6)

First-order, second-order differentials of t time, we have:

(.) = BU

dtds

dsd

dtd )(.)()( (A2.7)

(..) = 2

2

2

2

2

2

')'(.)()(BU

dtds

dsd

dtd

A2.8)

Replacing eqs.(A2.7), (A2.8) into eqs.(A2.3), (A2.4), then dividing eq.(A2.3) by BU /2

and eq.(A2.4) by 22 / BU , we have:

]''[2

'.2" **3

2*2

*1

2

22

HKKA

BhKH

mBh

UBh

UBh

hh

hh (A2.9)

]''[2

.2" *3

2*2

*1

4

2

22'

AKKA

BhKA

IB

UB

UB

(A2.10)

Page 7: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

7 | 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

Putting ,U

BK hh

UBK

, replacing to eqs.(A2.9), (A2.10):

]''[2

2" *3

2*2

*1

22'

HKKHBhKH

mB

BhK

BhK

Bh

hhh (A2.11)

*3

2'*2

'*1

42'

22" AKKA

BhKA

IBKK (A2.12)

Solution forms of the eqs.(A2.11), (A2.12) can be expressed under such ones as

follows:

h = h0 exp(it ) = h0 exp( )exp(). 0 iKshUsB

BiKU

(A2.13)

)exp()exp( 00 iKsti (A2.14)

Replacing eqs. (A2.13), (A2.14) into eqs. (A2.11), (A2.12), we have:

0*1

222

]2

2[ hHB

iKmBK

BKKi

BK

hhh

+ 0]22

[ 0*3

22

*2

22

HK

mBHKi

mB

0]2

2[]2

[ 0*3

22

*2

24

220

*1

24

HKmBAKi

IBKKKiKhA

BKi

IB

Conditioning that above homogenous equations have non-trivial solutions is that its

determinant must be zero:

]2

2[1]2

[

]22

[1]2

2[

*3

24

*2

24

222*1

24

*3

22

*2

22

*1

2222

AKIBAiK

IBKKKiK

BAiK

IB

HKmBHiK

mB

BHiK

mBKKKiK

Dethhh

=0 (A2.15)

Expanding the determinant (A2.15) and grouping by real part and imaginary one as

follow:

Det H = 21 i =0 A2.16)

Page 8: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

8 | 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

Placing hhK

KX

(A2.17)

The determinant (A2.15) is developed in such form as (A2.16). Then dividing the

determinant H by2hK , we have:

iXAI

BXAI

BXiXAI

B

iXHmBXiH

mBiXiH

mBXiX

H

hh

h

)2

2()2

()2

(

)2

(2

)2

2()1(

*2

4

2

22*

3

422*

1

4

2*2

22*

3

22*

1

22

iqqiP

iqqippH

222121

12111211

(A2.18)

Where: 12

11 Xp ; 2*1

2

12 22 XH

mBXp h

; 2*1

2

21 2XA

IBp

; XHmBq *

3

2

11 2

;

2*3

2

12 2XH

mBq

; 2

22*

3

42

21 2 h

XAI

BXq ; 2*

2

4

22 22 XA

IBq

h

Expanding determinant H to the real and imaginary parts, we have:

Real part: 0)( 122122122111 qpqpqp (A2.19)

Imaginary part: 0)( 112121122211 qpqpqp (A2.20)

With the real part (A2.19), we have following equation:

0)2

)(2

(

)2

2(2

22

)1(

2*2

22*

1

4

2*2

42*

1

2

2

22*

3

422

XHmBXA

IB

XAI

BXHmBXXA

IBXX

hhh

h

Developing and grouping by X, we have 1:

Page 9: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

9 | 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

]442

1[ *2

*1

42*1

*2

42*3

44

1 HAI

BmBHA

IB

mBA

IBX

+ ]2

22

.2[ *2

4*1

23 A

IBH

mBX h

h

+ ]

214[ *

3

4

2

22 A

IBX

hh

h

+ 2)(h

= 0 (A2.21)

Similarly developing to the imaginary part (A2.20), we have 2:

]4

.4

.22

[ *3

*1

42*3

*1

42*1

2*2

43

2 HAI

BmBAH

IB

mBH

mBA

IBX

+ ]2

222[ *3

42 A

IBX hh

h

+ ]

22[ *

2

4

2

2*1

2

AI

BHmBX

h

+

+ ]22[ 2

2

hhh

=0 (A2.22)

As a result, the flutter motion differential equations of 2DOF heaving- torsional

system have been transformed to two polynomical equations with -variable (critical-

state circular frequency or flutter frequency). Flow chart of critical wind velocity

determination by complex eigenvalue problem for 2DOF heaving and torisional

motion system can be shown in underneath figure.

Page 10: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

10 | 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

4. STEP-BY-STEP PROBLEM FOR 2DOF HEAVING-TORSIONAL MOTION

EQUATION SYSTEM

For solving 2DOF heaving-torsional motion equations, there are two powerful

analytical methods: so-called the complex eigenvalue method [Simui&Scanlan(1976)]

and the step-by-step method [Matsumoto(1995)]. 2DOF heaving-torsional motion

system has be usually taken cases of unit structural length subjected to unit self-

controlled forces into consideration. The 2DOF heaving-torsional motion systems,

moreover, can be known in sectional model tests in wind tunnels.

Fig 1. The scheme for analytical methods of 2DOF heaving-torsional flutter problems

(1) Complex eigenvalue method [Simiu&Scanlan(1976)]

The complex eigenvalue analytical method is based on some principles: i) Using some

techniques as transform of time-dimensionless variable, finding solutions under

harmonic manner; ii) Transform differential equations into linear equation system

with consistent condition its determinant must be zero; iii) Expanding determinant

and grouping by real and imaginary parts that must be simultaneously zero; iv)

Crossing point of solutions’ curves to determine the critical state of flutter instability.

(2) Step-by-step method [Matsumoto(1995)]

In principle, the step-by-step method is based on the serial solving technique of two

heaving-torsional motion equations, solutions of the former equation are to

determine coupled aerodynamic forces subjected to the later equation. From

2DOF Flutter Problems

Complex Eigenvalue Method

Step-by-step Method

Flutter Analytical methods

Page 11: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

11 | 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

transformation process, there is torsional-branch or heaving-branch step-by-step

method. Because torsional-branch instability dominates in almost cases, thus

torsional-branch step-by-step analysis will be favorable to be much more applicable

in comparision with heaving-branch one.

Stepwise procedure for torsional-branch analysis can be briefly presented hereinafter

i) The heaving motion equation will be taken into first account in which torsional-

related coupled forces are considered as external oscillation, furthermore heaving

motion solutions are found dependant on torsional vibration parameters; ii)

Obtained heaving motion solutions will be transformed into torsional motion

equation, then its damping ratio (or logarithmic decrement) will be determined in this

torsional-branch; iii) Checking such a damping ratio based on increment of reduced

wind velocity to obtain a critical condition for torsional-branch flutter instability.

Though the complex eigenvalue method has been applied for a long in solving 2DOF

heaving-torsional motion system to determine certain critical wind velocity, but

difficulty to investigate relationship of system damping ratio, system frequency on

wind velocity, and inter-relation between flutter derivatives as well. The step-by-step

method is favorable to deal with the complex eigenvalue method’s limitation.

DOF heaving-torsional flutter equations

The flutter motion equations of 2DOF heaving-torsional system can be written as

follow:

se

se

MKCI

LKCm

(3.1)

Where: KCm ,, are mass, damping coefficient and stiffness, respectively

associated with heaving motion.

Page 12: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

12 | 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

KCI ,, are mass inetia moment, damping coefficient and stiffness,

respectively associated with torsional motion.

Lse, Mse are self-controlled lift and moment.

The self-controlled forces Lse, Mse can be determined by either of Theodorsen’s

circulation function or Scanlan’s flutter derivatives under frequency approach. The

Scanlan’s self-controlled forces have been applied for the flutter motion equations for

various types of cross sections thank to experimentally-determined flutter

derivatives. Under this approach, the self-controlled forces per unit span length can

be modeled as follow:

b

kHkkHkUbkkH

UkkHUbL se

)()()()()2(

21 *

42*

32*

2*1

2

(3.2.a)

b

kHkkAkUbkkA

UkkAUbM se

)()()()()2(

21 *

42*

32*

2*1

22

(3.2.b)

Where: *3

*2

*4

*1 ,,, AAHH : uncoupled derivatives

*4

*1

*3

*2 ,,, AAHH : coupled derivatives

k is reduced frequency, U

bk

Above equations can be rewritten under standard form as follow:

b

kHkkHkUbkkH

UkkHUb

m

)()()()()2(212 *

42*

32*

2*1

22

b

kHkkAkUbkkA

UkkAUb

I

)()()()()2(212 *

42*

32*

2*1

222

(3.3.a); (3.3.b)

Where:

Page 13: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

13 | 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

, are free damping ratio and free circular frequency of heaving

motion, respectively

, are free damping ratio and free circular frequency of torsional

motion, respectively

IKC

mKC

IK

mK

2;

2

; 22

The step-by-step analysis method

Step 1: Free vibration parameters of single heaving and torsional motions

Free vibration parameters will be determined by free vibration equations:

0

0

KCI

KCm

(3.4)

Writing free vibration equations above under standard form:

02 2 (3.5)

02 2

Free vibration parameters are obtained as following

IKC

mKC

IK

mK

2;

2

; 22

(3.6)

Step 2: Solving the heaving motion equation in relation of coupled forces

Heaving motion equation can be written under expanded form as follow:

][][2 *3

23

*2

3*4

22

*1

22

H

mbH

mbH

mbH

mb

FFFF

Page 14: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

14 | 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[ *3

23

*2

3*4

22

2*1

2

HmbH

mbH

mbH

mb

FFFF (3.7)

Rewriting eq(3.7) in the standard form:

][2 *3

23

*2

32***

H

mbH

mb

FF (3.8)

Where: ** , are system circular frequency and system damping ratio of heaving

motion, respectively.

][ *4

22

22* Hmb

F

(3.9)

][2

2

*4

22

2

*1

2

*

Hmb

Hmb

F

F

(3.10)

Then, in order to transform the tortional-coordinate-related coupled forces in the

right-hand side to be pure external forces, technique for replacing the function can be

applied. Torsional displacement can be written under sinusoidal functional form:

t sin

)90sin(cos 0 tt (3.11)

Replacing the in to the coupled forces in left-hand side, we have:

2***2 tH

mbtH

mb

FF

sin)90sin( *

32

30*

2

3

(3.12)

Solution of eq(3.12) consists of such following components:

210 (3.13)

Where:

0 is total solution of free vibration equation:

Page 15: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

15 | 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

022*** (3.14)

1 is () solution of forced vibration equation:

)90sin(2 0*2

32*** tH

mb

F

(3.15)

2 is () solution of equation:

tHmb

F

sin2 *3

23

2*** (3.16)

Step 3: Finding solutions 1 , 2 of heaving forced vibration equation

(i) Finding 0 -solution: 02 12*

1**

1

We find 0 -solution under such a form: te t *

00 sin*

(3.17)

However, because system is motionless at initial time, thus solution of free vibration

is eliminated.

(ii) Finding 1 -solution: )90sin(2 0*2

3

12*

1**

1 tHmb

F

We find 1 -solution under such a form:

011 90sin t (3.18)

)(4)1(4)(2*

22*2

2*

22*

*2

3

22*2*222*

*2

3

1

H

mbH

mb

FF (3.19)

)2

(tan22*

**1

(3.20)

For convenience, we rewrite as follow:

Page 16: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

16 | 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

)(4)1(

||

2*

22*2

2*

22*

*2

3

1

H

mb

F (3.21)

111 sin t in cases of a) 01 90 when 0*

2 H (3.22)

b) 01 90 when 0*

2 H

(iii) Finding 2 -solution: tHmb

F

sin2 *3

23

22*

2**

2

We also find 2 -solution under such a form:

222 sin t (3.23)

)(4)1(4)(2*

22*2

2*

22*

*3

23

22*2*222*

*3

23

2

H

mbH

mb

FF (3.24)

)2

(tan22*

**1

For convenience, we rewrite as follow:

)(4)1(

||

2*

22*2

2*

22*

*3

23

2

H

mb

F (3.25)

222 sin t in cases of a) 2 when 0*3 H (3.26)

b) 02 180 when 0*

3 H

Thus, solution of heaving motion equation will be expressed:

)sin()sin( 221121 tt

Page 17: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

17 | 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

)cos()cos( 221121 tt

Expanding , and noting that tsin and

tcos , we have:

22221111

22221111

221121

sincossincos

cossincossincossincossin

)sin()sin(

tttt

tt

(3.27)

22221111

22221111

22221111

221121

sincossincos

sincossincos

sinsincoscossinsincoscos

)cos()cos(

tttt

tt

(3.28)

Step 4: Solving torsional motion equation

We have the torsional motion equation:

][][2 *4

24

*1

4*3

24

*2

42

A

IbA

IbA

IbA

Ib

FFFF (3.29)

Expanding the heaving-oriented forced excitation in right-hand side:

*

42

3*1

3

AIbA

Ib

FF

]sincossincos[ 22221111*1

3

AIb

F

]sincossincos[ 22221111*4

23

A

Ib

F

Page 18: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

18 | 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

)sin

cossincos(

)(4)1(

))((

2*3

*1

2

2*3

*11

*2

*1

21

*2

*1

2*

22*2

2*

22*

233

HA

HAHAHAmb

Ib

F

F

)sin

cossincos(

)(4)1(

))((

2*3

*4

2

2*3

*4

21

*2

*41

*2

*4

2*

22*2

2*

22*

233

HA

HAHAHAmb

Ib

F

FFF

F

])coscossinsin(

)sinsincoscos[(

)(4)1(

))()((

2*3

*4

21

*2

*42

*3

*11

*2

*1

2

2*3

*4

2

1*2

*42

*3

*11

*2

*1

2*

22*2

2*

2

2*

233

HAHAHAHA

HAHAHAHAmb

Ib

FFF

FFF

F

(3.30)

Replacing (3.30) in to eq(3.29), furthermore noting that in a torsional-branch

instability, the flutter frequency can be approximated to be torsional frequency, it

means that F (3.31)

][2 *3

24

*2

42

A

IbA

Ib

FF

])coscossinsin(

)sinsincoscos[(

)(4)1(

))()((

2*3

*4

21

*2

*4

22

*3

*1

21

*2

*1

2

2*3

*41

*2

*42

*3

*11

*2

*1

2*

22*2

2*

2

2*

233

HAHAHAHA

HAHAHAHAmb

Ib

FFFF

FFFF

FF

F

(3.32)

Page 19: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

19 | 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

Equation (3.32) can be rewritten under standard form:

022**

(3.33)

Where:

)sinsin

coscos(

)(4)1(

))()((21

)(21

2*3

*41

*2

*4

2*3

*11

*2

*1

2*

2*2

2*

2

2*

224

*2

4*

HAHA

HAHAmb

Ib

AIb

FF

FF

F

(3.34)

)coscos

sinsin(

)(4)1(

))()((

)(

2*3

*41

*2

*4

2*3

*11

*2

*1

2*

2*2

2*

2

22*

224

*3

24

22*

HAHA

HAHAmb

Ib

AIb

FF

FF

F

(3.35)

Solution of eq(3.33) can be expressed in such form: )sin( ** *

te t

in which: 2*2**

Step 5: Finding the critical condition of torsional instability

Flutter instability occurs if only if damping ratio 0*

0)sinsin

coscos(

)(4)1(

))()((21

)(21

2*3

*41

*2

*4

2*3

*11

*2

*1

2*

2*2

2*

2

2*

224

*2

4*

HAHA

HAHAmb

Ib

AIb

FF

FF

F

Page 20: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

20 | 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

(3.35)

Logarithmic decrement (Log. dec) ** 2

0)sinsin

coscos(

)(4)1(

))()((

)(

2*3

*41

*2

*4

2*3

*11

*2

*1

2*

2*2

2*

2

2*

224

*2

4*

HAHA

HAHAmb

Ib

AIb

FF

FF

F

1 = )(4

Ib

2 =)(4)1(

))((

2*

2*2

2*

2

2*

22

FF

F

mb

0)sinsincoscos(211 2*3

*41

*2

*42

*3

*11

*2

*1

*2

* HAHAHAHAA FF

We have: F , thus flutter condition can be rewritten as follow:

0)sinsincoscos(211 2*3

*41

*2

*42

*3

*11

*2

*1

*2

* HAHAHAHAA

Analytical procedure of step-by-step method

(1) Structural and dynamic parameters

- Structural parameters: b (2B

), KK ,

- Air density:

- Dynamic parameters: m, I, CC ,

(2) Flutter derivatives vs. reduced velocity bf

UUF

re

- Heaving derivatives: *4

*3

*2

*1 ,,, HHHH

Page 21: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

21 | 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

- Torsional derivatives: *4

*3

*2

*1 ,,, AAAA

(3) Free vibration characteristics

- Heaving motion:

mKC

mK

2;2

- Torsional motion:

IKC

IK

2;2

(4) Heaving-motion free vibration characteristics with uncoupled lift forces

][ *4

22

22* Hmb

F

][2

2

*4

22

2

*1

2

*

Hmb

Hmb

F

F

(5) Initial phase angle

)2

(tan22*

**1

case 0*2 H then 0

1 90 else 01 90

case 0*3 H then 0

2 180 else 2

(6) Torsional-branch circular frequency

)coscossinsin(211 2*3

*41

*2

*42

*3

*11

*2

*1

2*3

222* HAHAHAHAA FF

(7) Checking of torsional-branch log. dec

0)sinsincoscos(211 2*3

*41

*2

*42

*3

*11

*2

*1

*2

* HAHAHAHAA

Page 22: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

22 | 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 and dynamic input

CCKKImb ,,,,,,,

Free vibration parameters

,, ,

Wind velocity loop iU (Zero first approimation)

Circular frequency loop

jF , (First

,,,, ****ii AH

Frequency checking

jF ,*

End

UUU ii 1 jFjF ,1,

Log. Dec. checking 0j

Page 23: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

23 | 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

5. SINGLE-MODE, TWO-MODE AND MULTI-MODE FLUTTER

PROBLEMS OF NDOF SYSTEMS

As mentioned above, the conventional complex eigen method [Simui&Scanlan(1976)]

and the Step-by-step method [Matsumoto(1995)] are very powerful to solve for 1DOF

torsional flutter equation and 2DOF tortionnal-heaving equations of 2-dimensional

structures, some analytical methods have been developed for solving nDOF flutter

equations of 3-dimensional structures.

As first, happened flutter possibilities for bridge structures will be reviewed for

explanation of numerical analytical developments of nDOF flutter problems. By

various experiments and numerical analyses from practical applications of bridge

engineering, it is shown that the fundamental torsional vibration mode dominantly

involves to the flutter instability. Moreover, with bluff cross sections like low

slenderness ratio (B/D) rectangular sections or H-shape sections or stiffened truss

sections, the flutter instability almost occur in solely fundamental torsional mode

[Matsumoto (1996)] as known the torsional flutter as the case of Tacoma Narrow

failure. Whereas the fundamental torsional mode and any first symmetric or

asymmetric heaving mode usually couple mechanically at single frequency with the

streamlined cross sections as known as the coupled flutter or the classical flutter

(studied previously on aerodynamics of airplane’s airfoil wings). It is very

interesting, however, by both analyses and experiments to mark that coupled flutter

has occurred in case of the Akashi-Kaikyo bridge, that has been never seen before

with such kinds of the stiffen truss-girder cable-supported bridges [Katsuchi(1998)]. It

is questionable from case of the Akashi-Kaikyo bridge, thus, that coupled flutter also

possibly happens to very flexible long-span cable-supported bridges with bluff-

sections.

Page 24: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

24 | 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

Some recent analytical studies [Scanlan(1990), Pleif(1995), Jain(1996), Katsuchi(1998)],

furthermore, pointed out that in many cases of coupled flutter there are not the

fundamental torsional and heaving modes involved to the critical condition, but

many modes (multi-modes) superpose to gain more critical condition at a lower onset

velocity.

Some analytical methods for flutter problems have been developed from above

happened possibilities as single-mode [Simui&Scanlan(1976)], two-mode

[Scanlan(1981)], multi-mode methods [Scanlan(1990), Jain(1996), Katsuchi(1998),

Ge(2000)]. In principle, above-mentioned analytical methods are carried out on the

modal space thank to generalized coordinate transform and modal superposition

technique.

Diagram for analytical analysis methods of nDOF flutter problems

It is suggested that the single-mode method can be applied for cases of torsional

flutter possibly happens, whereas, the two-mode method for simplified approach and

nDOF Flutter problems

Multi-mode Method

Two-mode Method

Single-mode Method Torsional Flutter Heaving Flutter

Coupled Flutter

Bluff cross-sections Low B/D or H/- sections

Stiffen truss sections

Airfoil Thin plate sections

Streamlined sections

Page 25: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

25 | 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

the multi-mode method for more accuracy should be applied for tendency cases of

coupled flutter.

NDOF flutter motion equations

The motion equations of the nDOF structural systems in the steady flow can be

expressed under the Finite Element Method (FEM) as follow:

)(tPKUUCUM (A3.1)

Where: P(t) is the self-controlled flutter forces subjected to structure (However,

noting that the self-controlled flutter forces above are only valid in cases of

steady wind flows, whereas unsteady buffeting forces must be associated with

in self-controlled forces in the unsteady wind flows).

The self-controlled aerodynamic forces per unit length of bridge deck can are

popularly formed thank to the Scanlan’s experimentally-determined flutter

derivatives:

B

KHKKHKUBKKH

UKKHBUL se

)()()()(

21 *

42*

32*

2*1

2

BpKPKKPK

UBKKP

UpKKPBUD se )()()()(

21 *

42*

32*

2*

12

(A3.2)

B

KHKKAKUBKKA

UKKAUBM se

)()()()(

21 *

42*

32*

2*1

22

Where: *4

*1

*3

*2

*4

*1 ,,,,, PPAAHH : uncoupled derivatives

*3

*2

*4

*1

*3

*2 ,,,,, PPAAHH : coupled derivatives

Page 26: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

26 | 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

Above self-controlled aerodynamic forces can be explicitly divided by the

displacement-dependant aerodynamic elastic force-component and first-order

derivative-dependant aerodynamic damping one, we have:

UPUPtPtPtP 2121 )()()( (A3.3)

Thus, the nDOF flutter motion equations are written hereafter:

UPUPKUUCUM 21

0][][ 21 UPKUPCUM

0** UKUCUM (A3.4)

Where: 1* PCC ; 2

* PKK

C*, K* are the system damping-force and elastic-force matrices, respectively.

Because above matrices have no longer symmetrical, thus eigenvalues of

frequency equation of eq.(A3.4) must be conjugate complex pairs.

Transforming to the modal space and generalized coordinates

The motion equations can be transformed from the ordinary coordinates into the

principle ones (generalized coordinates) with the generalized coordinates are defined

as follow: U (A3.5)

Where: is the generalized coordinates

is the mass-normalized eigenmodal matrix

By using the mass-normalized technique, we transform

0** KCM

0**

KCI (A3.6)

Where: **CC T ; **

KK T

Page 27: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

27 | 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

Solution of eq.(A3.) found under such form: te (A3.7)

Expending and transforming eq.(A3.6) in the frequency equation (the

characteristic equation):

0**2 KCIDet (A3.8)

2n eigenvalues and eigenvectors can be determined through above equation.

Because the system damping and elastic matrices have no longer to be symmetrical,

thus eigenvalues from eq.(A3.8) will be exhibited under the conjugate complex

eigenvalue pairs:

iii j (A3.9)

Global response of bridge in the generalized coordinates can be obtained by superposing of combined modal responses as form:

n

i

tii

ie2

1

(A3.10)

Where: i is the modal scaling factor

n is the number of modes combined in global response

i is the eigenvector of ith mode, also presented under conjugate

complex eigenvector as iii jqp

Global response of bridge in the generalized coordinates can be rewritten hereby:

n

i

tjiiii

tjjiii

iiii ejqpjejqpj2

1

)()()(

Page 28: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

28 | 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

n

iiiiiiiiiii

t tpqtqpe i2

1cos2sin2

(A3.11)

Global response amplitude of bridge in the conventional coordinates can be expressed as follow:

N

iiiiiiiiiii

ti tpqtqpeU i

2

1]cos})}{{}}{({2sin})}{{}}{({2[}{}{

From eq.(A3.11), it can be seen clearly the role of the real part i of complex

eigenvalues in the system stability and instability problem, when real part of complex

eigenvalue become positive, system response amplitude is to be divergent and flutter

instability occurs (known as the Liapunov’s Theorem).

Linearly-discretized technique of the self-controlled aerodynamic forces

Uniform aerodynamic forces are linearly lumped at deck nodes. Six nodal displacements and their first derivatives can be expressed in to element coordinates as follows: TphU }000{}{ and TphU }000{}{ .

Element coordinates and nodal displacements

Page 29: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

29 | 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

The self-controlled aerodynamic forces along bridge deck can be linearly discretized

at any bridge deck node:

Diagram for nodal linear-discretization of self-controlled forces

From this linearly-discretized technique, finite-element damping and elastic

aerodynamic force matrices P1, P2 (12x12) can be easily obtained:

000000000000000000000000000000

41

*2

2*1

*2

*1

*2

*1

21

ABBABPPBHH

LUKBUP

000000000000000000000000000000000

41

*3

*3

*3

222

BAPH

LBKUP (A3.12)

Page 30: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

30 | 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

Noting that the matrices 21, PP (6x6) above are only presented at single node of

element, and element force matrices P1, P2 (12x12) will be built symmetrically from

above 21, PP (6x6).

Multi-mode flutter analysis

The quadratic eigenvalue problem will be difficult, in order to transform the

frequency equation (A3.8) into the standardized eigenvalue problem, motion

equation (A3.6) will be written in the state-space as follow:

00

000

**

KI

CII

(A3.13)

We replace hereby:

*

0CII

A ;

*

00

KI

B (A3.14)

teY

;

teY

We will have:

YBYA

BA

ZAZB (A3.15)

Here

Z

Expanding from eq.(A3.15), we have:

ZZBA 1

Page 31: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

31 | 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

ZZ

I

KC

0

**

Replacing:

0

**

I

KCD (A3.16)

As a result, the standardized eigenvalue problem has been achieved:

ZZD (A3.17)

The standardized eigenvalue problem above can be solved by the many solving

techniques such as Jacobi diagonalization, QL or QR transformation, subspace

iteration and another.

In general, the multi-mode method has been still based on prior selection of concrete

modes in combination. Recently, it can be automatically combined total modes from

free vibration analysis for flutter analysis, so-called full-mode method [Ge(2002)],

however, this full-mode method don’t pay much more accuracy out of control than

multi-mode method but time-consuming.

Single-mode flutter analysis in the modal space

The nDOF flutter motion equations can be written in the modal space and

generalized coordinate under such mass-normalized form:

21 PPKCI TT (A3.18)

Where: MI T ; CC T ; KK T

I is the mass-normalized unit matrix

C is the diagonal normalized damping matrix

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32 | 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

(containing modal damping coefficients)

K is the diagonal normalized elastic matrix

(containing modal eigenvalues)

Flutter motion equation of ith mode in the generalized coordinates can be written after

the normalized technique: )(2 2 tpiiiiiii (A3.19)

Where: pi(t) is the self-controlled aerodynamic force of ith mode (or called as

the normalized generalized aerodynamic force) determined as follow:

iTii

Tii PPtp 21)( (A3.20)

and i =

(x) (x)(x)(x)p(x)(x)

ji

ji

ji

orpor

or

(A3.21)

x is an deck-alongside coordinate

i, j are an index for combination between two modes

Expanding (A3.20) with noting that aerodynamic matrices P1, P2 determined by

eq.(A3.12) and grouping in generalized coordinates ( ) and their first-order

derivatives ( ), the normalized generalized aerodynamic force can be obtained

below:

ihhppphhhi jijijijijijiGABGBAGBPGPGBHGH

UBKUtp

][21)( *

22*

1*

2*

1*2

*1

2

iph jijijiGBAGPGHBKU ][

21 *

3*

3*3

22 (A3.22)

Where: Grmsn is the modal integral sums determined by such formula in

discretized manner of mode shapes.

Page 33: Flutter stability analysisuet.vnu.edu.vn/~thle/Flutter stability analysis.pdfFLUTTER STABILITY ANALYSIS THEORY AND EXAMPLE Prepared by Le Thai Hoa 2004 2 | L e T h a i H o a – F

33 | 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

Grmsn =

m

1kkl (r,k)m (r,k)n (A3.23)

Omitting cross-modal integral sums Grmsn (rs) due to their small, remaining auto-

modal integral ones Grmsn (r=s), we easily obtain:

ipphhi jijijiGABGPGH

UBKUtp

][21)( *

22*

1*1

2 ijiGBABKU ][

21 *

322 (A3.24)

Putting eq.(A3.24) in eq.(A3.19), transforming into an ordinary 1DOF differential

equation:

0)](21[)](

212[ *

3222*

22*

1*1

2 iiipphhiii jijijijiGBABKUGABGPGHU

02 iiiiii (A3.25)

Where:

jiGKAB

i

ii

)(2

1 *3

4

22

(A3.26)

i = jii*2

2pipji

*1hihji

*1

4

i

ii )GK(AB)GK(PG)K([H4

Bαα

ρωω

(A3.27)

U

BK ii

(A3.28)

Two-mode flutter analysis in the modal space

In cases of two-mode coupled flutter, two modes as common sense are engaged: one is

dominant heaving mode and other is dominant torsional mode. It might be postulated

that two modes can couple if they have similarly modal shapes and closely natural

frequencies, and cross-modal integral sums in these cases play more important role.

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34 | 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

Generally, the two-mode flutter method has developed from problems of nDOF system’s

single-mode analysis and of 2DOF system’s complex eigen analysis. The two motion

equations of ith and jth modes with the coupled normalized generalized aerodynamic

forces can be expressed following:

i) ith modal motion equation

iiiiii 22

ihhppphhh iiiiiiiiiiiiGABGBAGBPGPGBHGH

UBKU

][21 *

22*

1*

2*

1*2

*1

2

ihppphhh iiiiiiiiiiiiGABGBAGBPGPGBHGHKU ][

21 *

32*

4*

3*

4*3

*4

22

+ jhhppphhh jijijijijijiGABGBAGBPGPGBHGH

UBKU

][21 *

22*

1*

2*

1*2

*1

2

jhppphhh jijijijijijiGABGBAGBPGPGBHGHKU ][

21 *

32*

4*

3*

4*3

*4

22 (A 3.29a)

ii) jth modal motion equation

jjjjjj 22

jhhppphhh jjjjjjjjjjjjGABGBAGBPGPGBHGH

UBKU

][21 *

22*

1*

2*

1*2

*1

2

jhppphhh jjjjjjjjjjjjGABGBAGBPGPGBHGHKU ][

21 *

32*

4*

3*

4*3

*4

22

ihhppphhh ijijijijijijGABGBAGBPGPGBHGH

UBKU

][21 *

22*

1*

2*

1*2

*1

2

ihppphhh ijijijijijijGABGBAGBPGPGBHGHKU ][

21 *

32*

4*

3*

4*3

*4

22 (A 3.29b)

Solution for two modal motion equations under coupled forces can be carried out by

following steps:

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35 | 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

Step 1: Solutions assumed under such forms as tiii

Fe 0 , tijj

Fe 0 . Then expanding

and grouping into equation system under terms of 00 , ji . We can obtain 2 equations:

iT

ii

F

i i

)](21)[( 2

][{21 *

22*

1*

2*

1*2

*1

2iiiiiiiiiiii

GABGBAGBPGPGBHGHiB hhppphhh

0*3

2*4

*3

*4

*3

*4 ]}[ ihppphhh iiiiiiiiiiii

GABGBAGBPGPGBHGH

][{21 *

22*

1*

2*

1*2

*1

2jijijijijiji

GABGBAGBPGPGBHGHiB hhppphhh (A 3.30a)

0*3

2*4

*3

*4

*3

*4 ]}[ jhppphhh jijijijijiji

GABGBAGBPGPGBHGH

and

jT

jj

F

j i

)](21)[( 2

][{21 *

22*

1*

2*

1*2

*1

2jjjjjjjjjjjj

GABGBAGBPGPGBHGHiB hhppphhh

0*3

2*4

*3

*4

*3

*4 ]}[ jhppphhh ijjjjjjjjjjj

GABGBAGBPGPGBHGH

][{21 *

22*

1*

2*

1*2

*1

2ijijijijjjjj

GABGBAGBPGPGBGHiB hhppphhh (A 3.30b)

0*3

2*4

*3

*4

*3

*4 ]}[ ihppphhh ijijijijjjij

GABGBAGBPGPGBHGH

Step 2: Grouping by 00 , ji , obtaining two-equation system, conditioning nontrivial

solutions that their determinant must be zero. We can write the determinant under such a

form:

22222121

12121111

iBAiBAiBAiBA

Det

(A 3.31)

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36 | 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

Step 3: Expanding the determinant, two equations of real and imaginary parts can be

obtained and must be simultaneously zero, we have:

Real part: 0 ijjiijjijjiijjii BBAABBAA (A 3.32a)

Imaginary part: 0 ijjiijjijjiijjii ABBAABBA (A 3.32b)

Where:

])[(2/11)/( *3

2*4

*3

*4

*3

*4

22iiiiiiiiiiii

GABGBAGBPGPGBHGHBA hppphhhFiii

])[(2/1 *3

2*4

*3

*4

*3

*4

2jijijijijiji

GABGBAGBPGPGBHGHBA hppphhhij

])[(2/11)/( *3

2*4

*3

*4

*3

*4

22jjjjjjjjjjjj

GABGBAGBPGPGBHGHBA hppphhhFjjj

])[(2/1 *3

2*4

*3

*4

*3

*4

2ijijijijjjij

GABGBAGBPGPGBHGHBA hppphhhji

])[(2/1)/(2 *2

2*1

*2

*1

*2

*1

2iiiiiiiiiiii

GABGBAGBPGPGBHGHBB hhppphhhFiiii

])[(2/1 *2

2*1

*2

*1

*2

*1

2jijijijijiji

GABGBAGBPGPGBHGHBB hhppphhhij

])[(2/1)/(2 *2

2*1

*2

*1

*2

*1

2jjjjjjjjjjjj

GABGBAGBPGPGBHGHBB hhppphhhFjjjj

])[(2/1 *2

2*1

*2

*1

*2

*1

2ijijijijijij

GABGBAGBPGPGBHGHBB hhppphhhji

Step 4: Solutions of Eq.(A 3.32a), Eq.(A 3.32b) are found simultaneously, intersected point

of solution curves determine the critical flutter condition.

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37 | 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

6. NUMERICAL EXAMPLE OF A CABLE-STAYED BRIDGE

Numerical example of a cable-stayed bridge for the flutter analysis will be presented

under three approaches:

i. Complex-eigen analysis for a 2DOF torsional-heaving system (first

torsional and heaving modes selected for the analysis)

ii. Step-by-step analysis for a 2DOF torsional-heaving system ( also first

torsional and heaving modes selected for the analysis)

iii. Single-mode and multi-mode analysis for the cable-stayed bridge

Fig A4.1. Cable-stayed bridge for numerical analysis example

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38 | 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 characteristics and flutter derivatives

Table A4.1. Sectional characteristics of example cable-stayed bridge

Gider Tower Stayed cables Material parameters E =3600000 T/m2 G =1384600 T/m2 =0.3 Poison ratio Geometrical parameter A =6.525 m2 I33 =0.11 m4 I22 =114.32 m4 J =0.44m4

Material parameters E =3600000 T/m2 G =1384600 T/m2 =0.3 Poison ratio Geometrical parameter A =1.14 m2; I33=0.257 m4 I22 =0.118 m4;J=0.223m4 A =1.14 m2; I33=0.257 m4 I22 =0.118 m4;J =0.223m4

Material parameters E = 19500000 T/m2 Geometrical parameter A =26.355 cm2 Type 19K15 A =16.69 cm2 Type 12K15

Fig A4.2. Diagrams of the flutter derivatives H*i, A*i (i=1-3) given by

quasi-steady formula [Scanlan(1989), Pleif(1995)]

Flutter derivatives

-20

-15

-10

-5

0

5

10

15

20

0 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2

Reduced velocity K

Hi

(i=1,2

,3)

Flutter derivatives

-0.5

0

0.5

1

1.5

2

2.5

3

3.5

0 1 2 3 4 5 6 7 8 9 10 11 12

Reduced frequency K

A*i

(i=

1,2,

3)

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39 | 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

Free vibration analysis

Mode 1 f=0.6099Hz

Mode 2 f=0.8016Hz

Mode 3 f=0.8522Hz

Mode 4 f= 1.1949Hz

Mode 5 f=1.2931Hz

Mode 6 f=1.4495Hz

Mode 7 f=1.5819Hz

Mode 8 f=1.6304Hz

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40 | 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

Fig A4.3. Fundamental 10 natural mode shapes

Free vibration characteristics and modal integrals

Table A4.1. Characteristics of the first 10 natural mode shapes

Mode Eigenvalue Frequency Period Modal Character

shape 2 (Hz) (s)

1 1.47E+01 0.609913 1.639579 S-V-1

2 2.54E+01 0.801663 1.247406 A-V-2

3 2.87E+01 0.852593 1.172893 S-T-1

4 5.64E+01 1.194920 0.836876 A-T-2

5 6.60E+01 1.293130 0.773318 S-V-3

6 8.30E+01 1.449593 0.689849 A-V-4

7 9.88E+01 1.581915 0.632145 S-T-P-3

8 1.05E+02 1.630459 0.613324 S-V-5

9 1.12E+02 1.683362 0.594049 A-V-6

10 1.36E+02 1.857597 0.53830 S-V-7

Note : S: Symmetric mode V: Heaving mode shape

A: Asymmetric mode T: Torsional mode shape

P: Horizontal mode shape

Mode 7 f=1.5819Hz

Mode 8 f=1.6304Hz

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41 | 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

D¹ng dao ®éng thø 2 ( D¹ng uèn thø 2)

-0,15

-0,1

-0,05

0

0,05

0,1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Gi¸

trÞ d

¹ng

D¹ng dao ®éng thø 3 (D¹ng xo¾n thø nhÊt )

-0,02

-0,015

-0,01

-0,005

0

0,005

0,01

0,015

0,02

1 4 7 10 13 16 19 22 25 28

Gi¸

trÞ d

¹ng

D¹ng dao ®éng thø 4 ( d¹ng xo¾n thø 2 )

-0,02

-0,015

-0,01

-0,005

0

0,005

0,01

0,015

1 4 7 10 13 16 19 22 25 28

Gi¸

trÞ d

¹ng

D¹ng dao ®éng thø 5

(d¹ng uèn thø 3)

-0,12

-0,1

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Gi¸

trÞ d

¹ng

D¹ng dao déng thø 6(d¹ng uèn thø 4)

-0,15

-0,1

-0,05

0

0,05

0,1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Gi¸

trÞ d

¹ng

D¹ng dao ®éng thø 7

(d¹ng xo¾n thø 3)

-2,00E-02

-1,50E-02

-1,00E-02

-5,00E-03

0,00E+00

5,00E-03

1,00E-02

1 4 7

10 13 16 19 22 25 28

Gi¸

trÞ d

¹ng

D¹ng dao ®éng thø 8(d¹ng uèn thø 4)

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,06

0,08

0,1

0,12

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29Gi¸

trÞ d

¹ng

Fig. Modal amplitude of normalized mode shapes

D¹ng dao ®éng thø nhÊt (D¹ng uèn thø nhÊt )

-0,12

-0,1

-0,08

-0,06

-0,04

-0,02

0

0,02

0,04

0,061 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Gi¸

trÞ d

¹ng

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42 | 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

Table A4.2. Modal integral sums of first 10 natural mode shapes

Mode Frequency Modal Modal integral sums Grmsn

shape (Hz) Character Ghihi Gpipi Gii

1 0.609913 S-V-1 5.20E-01 7.50E-11 0.00E+00

2 0.801663 A-V-2 4.95E-01 7.43E-09 1.35E-09

3 0.852593 S-T-1 3.79E-09 5.23E-05 1.14E-02

4 1.194920 A-T-2 1.78E-07 1.82E-05 1.07E-02

5 1.293130 S-V-3 5.07E-01 1.36E-07 23.62E-09

6 1.449593 A-V-4 4.99E-01 2.10E-09 9.42E-09

7 1.581915 S-T-P-3 2.67E-07 1.10E-03 1.10E-02

8 1.630459 S-V-5 5.03E-01 1.43E-07 1.27E-08

9 1.683362 A-V-6 1.64E-06 1.77E-04 1.09E-02

10 1.857597 S-V-7 4.16E-06 2.78E-03 1.11E-02

Note: Modal integral sums: Grmsn =

m

1kkl (r,k)m (r,k)n

lk: Discretized deck lengths (r,k)n : Discretized modal values

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43 | 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

FigA4.4.Diagram of complex eigen solutions of 2DOF torsional-heaving system (first

heaving mode + first torsional mode)

1 2 3 4 5 6 7 8-1.5

-1

-0.5

0

0.5

1

1.5

2

Reduced Frequency K

X33

X44

X43

X32

X41 X31 X 42

Intersection

5.3

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44 | 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

Fig.A4.5.Diagram of wind velocity vs. system damping ratio (V-)

Fig.A4.6.Diagram of wind velocity vs. frequency (V-f)

10 20 30 40 50 60 70 80 900.6

0.7

0.8

0.9

1

1.1

1.2

1.3

Wind velocity (m/s)

Freq

uenc

y (H

z)

Mode 3 (Torsional)Mode 4 (Torsional)

Mode 3

Mode 4

Aerodynamic interaction

Aerodynamic interaction

10 20 30 40 50 60 70 80 90-0.2

0

0.2

0.4

0.6

0.8

1

1.2

Wind velocity (m/s)

Sys

tem

dam

ping

ratio

Mode 1 (Heaving)Mode 2 (Heaving)Mode 3 (Torsional)Mode 4 (Torsional)Mode 5 (Heaving)

Mode 1 Mode 2

Mode 5

Mode 3

Mode 4

64.5 88.5

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45 | 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

Time-history modal amplitude of first 5 modes at certain wind velocities

Modal amplitude of first 5 modes at U=50m/s Modal amplitude of first 5 modes at U=65m/s

Modal amplitude of 5 modes at U=70m/s Modal amplitude of 5 modes at U=90m/s

Fig.A4.7.Diagrams of modal amplitudes of first 5 modes at some certain wind

velocity values

0 10 20 30 40 50 60 70 80 90 100-1

0

1Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 2

0 10 20 30 40 50 60 70 80 90 100-5

0

5 Mode 3

Mod

al A

mpl

itude

0 10 20 30 40 50 60 70 80 90 100-1

0

1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

Mode 4

0 10 20 30 40 50 60 70 80 90 100-1

0

1Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 2

0 10 20 30 40 50 60 70 80 90 100-1

0

1x 10

5

Mod

al A

mpl

itude

0 10 20 30 40 50 60 70 80 90 100-2

0

2 Mode 4 (Divergence)

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

Mode 3 (Divergence)

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 3

Mod

al A

mpl

itude

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 4

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

Mode 20 10 20 30 40 50 60 70 80 90 100

-1

0

1 Mode 1

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 2

0 10 20 30 40 50 60 70 80 90 100-2

0

2 Mode 3

Mod

al A

mpl

itude

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 4

0 10 20 30 40 50 60 70 80 90 100-1

0

1 Mode 5

Time (s)

(Divergence)

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46 | 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

Investigation on change of modal amplitude of some major modes following wind

velocities and time intervals

Diagram of 1st heaving modal amplitude vs. wind velocity after 2 seconds

Diagram of 2nd heaving modal amplitude vs. wind velocity after 2 seconds

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

-0.15

-0.1

-0.05

0

0.05

0.1

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

Decay

Decay

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47 | 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

Diagram of 1st torsional modal amplitude vs. wind velocity after 2 seconds

Diagram of 2nd torsional modal amplitude vs. wind velocity after 2 seconds

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

0.015

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

Divergence

Divergence

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48 | 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

Diagram of 3nd heaving modal amplitude vs. wind velocity after 2 seconds

Diagram of 1st heaving modal amplitude at wind velocity 50m/s vs. time intervals

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

Initial50m/s65m/s70m/s90m/s

Decay

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

(at 5

0m/s

)

Initial

1second

2seconds

3seconds

5seconds

10seconds

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49 | 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

Diagram of 1st heaving modal amplitude at wind velocity 70m/s vs. time intervals

Diagram of 1st torsional modal amplitude at wind velocity 50m/s vs. time intervals

-0.12

-0.1

-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

0.06

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

(at 7

0m/s

)

Initial

1second

2seconds

3seconds

5seconds

10seconds

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

(at 5

0m/s

)

Initial1second2seconds3seconds5seconds10seconds

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50 | 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

Diagram of 1st torsional modal amplitude at wind velocity 50m/s vs. time intervals

-0.025

-0.02

-0.015

-0.01

-0.005

0

0.005

0.01

1 3 5 7 9 11 13 15 17 19 21 23 25 27 29

Deck nodes

Mod

al a

mpl

itude

(at 7

0m/s

)

Initial1second2seconds3seconds5seconds10seconds