modeling overview and design implications of stay-cable vibrations: modeling overview and design...
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Damping of StayDamping of Stay--Cable Vibrations:Cable Vibrations:Modeling Overview and Design ImplicationsModeling Overview and Design Implications
Joseph A. MainJoseph A. MainNational Institute of Standards and TechnologyNational Institute of Standards and Technology
Nicholas P. JonesNicholas P. JonesJohns Hopkins UniversityJohns Hopkins University
2006 Wind Induced Vibrations of Cable Stay Bridges Workshop
April 26, 2006
• Taut string with viscous damper• Influence of bushings at anchorage• Influence of bending stiffness• Influence of damper nonlinearities• Concluding remarks
OutlineOutline
Taut string model:• Neglects flexural stiffness: “kink” at damper• Neglects sagged profile, axial extensibility• Reasonable approximation in many cases
Taut String with Viscous DamperTaut String with Viscous Damper
x c
T T
m
cxL
• Identified numerically• Relates modal damping ratios to viscous damper coefficient
• Useful in damper design for stay cables
“Universal Estimation Curve”“Universal Estimation Curve”(Pacheco, (Pacheco, FujinoFujino, and , and SulekhSulekh 1993)1993)
0
0.2
0.4
0.6
0 0.5 1 1.5 2
ˆ /ccnx L
/n
cx Lζ
ˆ cTm
c =
0c → c → ∞
• PDE over two segments:• Separation of variables:• BC’s and continuity:• Force balance at damper:
Transcendental equation for eigenfrequencies:
Analytical Solution for Taut StringAnalytical Solution for Taut String((KrenkKrenk 2000, Main and Jones 2002)2000, Main and Jones 2002)
x c
T T
m
cx
my Ty′′=&&
( , ) ( ) iy x t Y x e τΩ=
1T
L mπω
−⎛ ⎞
= Ω⎜ ⎟⎜ ⎟⎝ ⎠
L
[ ( ) ( )] ( )c c cT y x y x cy x+ −′ ′− = &
(0) 0, ( ) 0, ( ) ( )c cy y L y x y x+ −= = =
sin( ) sin sin 0c cx L xciL LTm
πω πω πω −⎛ ⎞ ⎛ ⎞+ =⎜ ⎟ ⎜ ⎟⎝ ⎠ ⎝ ⎠
ComplexComplex--Valued Valued EigenfrequenciesEigenfrequencies
0
0.02
0.04
0.5 1 1.5 2 2.5 3 3.5
Re[ ]ω
Im[ ]ωundamped limit
-1
0
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
“clamped” limit
-1
0
1
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
0.02cxL
=
• Real part: damped frequency• Imaginary part: decay rate• Damping ratio:• Locus in complex plane:
Im[ ] /ζ ω ω=
Complex-valued frequency increment:(assumed small)
Asymptotic ApproximationsAsymptotic Approximations0n n nω ω ωΔ = −
Asymptotic approximation:ˆ /optn cc n x Lπ≅ˆ ˆ/
ˆ ˆ1 /
optn
n n optn
ic cic c
ω ω∞Δ ≅ Δ+ /n cn x Lω∞Δ ≅
where
0
0.5
0 0.5 1
Im[ ]nωΔ
Re[ ]nωΔ0
12 nω ∞Δ
nω ∞Δ12 nω ∞Δ
0
0.5
0 1 2 3 4 5
ˆ ˆ/ optnc c
Im[ ]nωΔ
0
12 nω ∞Δ
increasing c
Semi-Circular Locus
Influence of Bushings at AnchorageInfluence of Bushings at Anchorage
ANCHORAGE BOX
GUIDE PIPE
ANCHOR HEAD
NEOPRENE RUBBER BUSHING
LOAD DISTRIBUTION PLATE
GROUT CAP
TENSION RING
Bushing Removal: Stay with DamperBushing Removal: Stay with Damper
Fred Hartman Bridge(Houston, Texas)
• Cable: taut string• Damper: linear dashpot• Bushing: linear spring• Boundary conditions, continuity at damper
Transcendental equation for complex eigenfrequencies– Efficient iterative solution scheme– Explicit asymptotic approximate solution
Analytical Model: Damper and BushingAnalytical Model: Damper and Bushing
kx
L
x
k c
T Tm
cx
my Ty′′=&&
[ ( ) ( )] ( )c c cT y x y x cy x+ −′ ′− = &
[ ( ) ( )] ( )k k kT y x y x ky x+ −′ ′− =
ComplexComplex--Valued Valued EigenfrequenciesEigenfrequencies
0
0.005
0.01
0.015
0 20 40 60 80 100 120 140 160 180 2000
0.005
0.01
0.015
0 0.005 0.01 0.015 0.02
0.001
0.01
0.1
1
10
100
1000
0 0.005 0.01 0.015 0.02
cRe[ ]nωΔ
increasing
Asymptotic:Exact:
0χ =0.5χ =2χ =10χ =
ˆ 0c =
ˆ( 0)cnω =Δ
Im[Δ
ωn]
Im[Δ
ωn]
Damper Coeficient,
Bushing Only
Frequency Shift,
Dec
ay R
ate,
Bush
ing
Stiff
ness
, χ
1n =
/ 0.75k cx x =
/ 0.02cx L =kkx
Tχ =
ˆ cTm
c =c
Generalization of “Universal Curve”Generalization of “Universal Curve”
0
0.2
0.4
0.6
0.8
1
0 5 10 15 20
effc
c
xx
/ 0.2k cx x =
χ
/ 0.4k cx x =
/ 0.6k cx x =
/ 0.8k cx x =
/ 1.0k cx x =0
0.2
0.4
0.6
0 0.5 1 1.5 2
/ˆ effc Lcnx
/n
effc Lxζ
1 (1 / )1
effc k c
c
x x xx
χχ
+ −=
+
2
ˆ /ˆ/ 1 ( / )
effn c
eff effc c
cnx Lx L cnx L
ζ ππ
≅+
Effective Damper Location: “Universal Curve”:
• Effective damper location:
• Generalized “universal curve”:
Bushing Stiffness,
Effects of Bushing StiffnessEffects of Bushing Stiffness
Reduction of leads to:• Reduction of attainable damping ratios:
• Increase of optimal damper coefficient:
12
effopt cn
xL
ζ ≅
/optn eff
c
Tmcnx Lπ
≅
effcx
• PDE over two segments:• Force balance at damper:• Boundary conditions, continuity at damper
Transcendental equations for complex eigenfrequencies– Efficient iterative solution schemes– Explicit asymptotic approximate solutions
Influence of Bending StiffnessInfluence of Bending Stiffness
T T
c ,m EI
L
cxTT
c ,m EI
Pinned supports:
Clamped supports:
2TLEI
γ =
10 600γ< <most stay cables:
• Tensioned beam with viscous damper:
0EIy Ty my′′′′ ′′− + =&&[ ( ) ( )] ( )c c cEI y x y x cy x− +′′′ ′′′− = &
nondimensionalparameter:
Asymptotic ApproximationsAsymptotic ApproximationsSame form as for taut string!
Semi-circular locus persists:
Expressions for and depend on support conditionsˆoptnc
ˆ ˆ/ˆ ˆ1 /
optn
n n optn
ic cic c
ω ω∞Δ ≅ Δ+
nω∞Δ
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1
n = 1n = 2n = 3Asymptotic
0
0.2
0.4
0.6
0 0.2 0.4 0.6 0.8 1
n = 1n = 2n = 3Asymptotic
Im[ ]n
n
ωω∞
ΔΔ
Re[ ] /n nω ω∞Δ ΔRe[ ] /n nω ω∞Δ Δ
Pinned supports: Clamped supports:
50γ =
/ 0.02cx L =
Evolution of “Universal Curve”Evolution of “Universal Curve”
Pinned supports: Clamped supports:
0
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 30
0.2
0.4
0.6
0.8
1
0 0.5 1 1.5 2 2.5 3
100010050
γ :100010050
γ:
/n
cx Lζ
/n
cx Lζ
/ 0.02cx L =
Shifting of curve depends on support conditions:
ˆ /ccnx L ˆ /ccnx L
Optimal Damping RatiosOptimal Damping Ratios
Pinned supports: Clamped supports:
0.1
1
10
0.1 1 10 100
0.005
0.01
0.02
0.05
0
0.2
0.4
0.6
0.8
1
0.1 1 10 100 1000 10000
0.0050.010.020.05
1
/
opt
cx Lζ
/cx Lγ γ
/ :cx L
1
/
opt
cx Lζ
10γ >
• Larger frequency shifts for pinned supportslarger damping
/ :cx L
Optimal Damper CoefficientOptimal Damper CoefficientPinned supports: Clamped supports:
0.1
1
10
0.1 1 10 100
0.005
0.01
0.02
0.05
0.1
1
10
100
0.1 1 10 100
0.0050.010.020.05ˆopt c
nxc nL
/cx Lγ
• Significant increases over “taut string” values (currently used in design)
• Much stronger increase for clamped supports• Within 20% of taut string values for
/ :cx L / :cx L
ˆopt cn
xc nL
/cx Lγ
/ 10cx Lγ >
Damper NonlinearitiesDamper Nonlinearities
Power-Law Damper:
dF β = 1
v
β > 1
β < 1
β ≈0
( ) sgn( )dF v c v vβ=( ) sgn( )d oF v F v c v= + ⋅
dF
oF
oF
c1
v
b)
Friction/Viscous Damper:
Equivalent Viscous SolutionsEquivalent Viscous Solutions• Compute coefficient of “equivalent viscous damper” based
on energy dissipation:
• Equivalent viscous coefficient depends on frequency of oscillation and amplitude at damper
2deq
Wc
πα ωΔ
=
Power-Law Damper
where
Friction/Viscous Damper1( ) ( )eqc c fβωα β−=
2 (1 / 2)( )(3 / 2 / 2)
f βββπ
Γ +=
Γ +
4 oeq
Fc cπωα
= +
αω
• Inaccurate for strong nonlinearities (e.g., frictional “stick-slip”) efficient numerical solution scheme developed
0
0.1
0.2
0.3
0.4
0.5
0.6
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
1
1.75
2.2
2.4
Friction/Viscous DamperFriction/Viscous Damper• Nondimensional friction/amplitude parameter:• Influence on “Universal Curve”:
– Reductions in optimal value of viscous coefficient– Eventually, reductions in attainable damping ratios
increasing μ
10F An
T Lμ
−⎛ ⎞= ⎜ ⎟⎝ ⎠
:μ
ˆ /ccnx L
/cx Lζ
• Damping independent of mode numberAdvantage: damp multiple modes
• Damping is a function of peak modal amplitude, A• Optimal amplitude can be designed (e.g., set Aopt = 1.5D)
SquareSquare--Root Damper (Root Damper (β β = = 0.50.5))
0
0.1
0.2
0.3
0.4
0.5
0.6
0 1 2 3 4 5 6 7 8 9 10
/ optA A
/cx Lζ
• Importance of damper-induced frequency shifts• Persistence of semi-circular locus and general form of
“universal curve”• Optimal damper coefficient and maximum attainable
damping ratios affected by bushings, bending stiffness• Asymptotic approximations obtained for many cases of
practical interest: efficient and useful in design• “Equivalent viscous” approach used to extend linear
solutions to nonlinear dampers– Friction can reduce optimal damper coefficient and damping ratios
should be accounted for in design– Damping of square-root damper independent of mode number
Concluding RemarksConcluding Remarks