flutter stability analysis - vnuthle/flutter stability analysis.pdfflutter stability analysis 1....
TRANSCRIPT
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:
][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.
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)
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)
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:
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.
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
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.
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:
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
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:
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:
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
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
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)
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
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
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
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
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.
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
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
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
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
)()()(
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
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)
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
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
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.
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.
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:
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)
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.
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
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)
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
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
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
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
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
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
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)
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
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
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
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
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