radiation damping

Faculty of Civil Engineering

Radiation damping caused by wavepropagation in Soil-Structure Interaction

( )(SSI)

Similar to Fluid-Structure Interaction (FSI)

Soil: unbounded domain

P bl D i b h i f ilProblem: Dynamic behaviour of soil(stiffness, damping)frequency-dependant

Peter Ruge

Content

• Three typical interaction problems

• Procedures based upon Codes

• Results from theory / computational dynamics- Dynamic stiffness/compliance with BEM- Dynamic stiffness analytically

• Frequency to Time transformation

- „Inverse“ Modal elimination

• Rational interpolation

• Asymptotic behaviour

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Train on infinite railway- viscoelastically restrained beam -viscoelastically restrained beam

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Content

Überschrift 1Text

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Procedure by codes

Rigid foundation; 6 DOF‘s 3 translational springs/dampers3 rotational springs/damperp g / p

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Soil springs, viscous damperCalculated numerically; semi analyticallyCalculated numerically; semi-analytically

G: Dynamical shear modulus [N/m2]G: Dynamical shear modulus [N/m2]ν: Poisson‘s ratioρ: Mass density [kg/m3]J: Rotation inertia (axis through mass center S of body)

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J: Rotation inertia (axis through mass-center S of body)

Results from theory / Computational Dynamics

Values k, d are frequency-dependent

K for harmonic situation

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Input

Output

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Stiffness from BEM

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Semi-infinite rod resting on elastic foundation

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Frequency to Time Transformation

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Modal Elimination

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Result of Elimination

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Rational interpolation of K(Ω)H d i it di tiHere procedure in opposite direction

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Least-squares approach. Matrix-valued

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Switching towards liner frequency representation

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Real interpolation for

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Splitting procedure; example

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Internal variables

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Asymptotic behaviour

C tl

Wave propagation in infinite space domainsdue to transient excitations

Consequently

Typical behaviour Typical behaviour

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For high frequency range: Shear deformations

Rotary inertia independent state variables

AnalyticalWave-type solution

independent state variables

yp


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for

Analytical formulationin frequency domainin frequency domain

Splitting into 2 parts

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time

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timea

t=0

0 1 2 k-23 k-1 k t

tkt 0

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Example:Infinite Timoshenko beam with momentumInfinite Timoshenko-beam with momentum-impact

5

6M = 5M = 7M = 9

2

3

4

[1

0-4 m

]

M = 9

0

1

2

Rot

atio

n φ

-2

-1

0 0.2 0.4 0.6 0.8 1

Time t [10-3 s]

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Time t [10-3 s]

Final remark

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