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

AcknowledgementsAcknowledgements

• TXDOT• FHWA• NSF• Jack Spangler, senior mechanical technician, JHU