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Dynamics of Structures 2019-2020 6. Remedial measures

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6. Remedial measures

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Dynamics of structures

Arnaud Deraemaeker ([email protected])

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A test-case based learning of vibrations in civil engineering

Case study 1 : pedestrian induced vibrations of a footbridge

• Source of excitation• Effects• Design methodology• Remedial measures


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Typical frequencies of footbridges

Vibration problems in structures, H. Bachman, 1995

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Design methodology : High vs low tuning

Low tuning High tuning


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5Vibration problems in structures, H. Bachman, 1995

Design methodology : Influence of stiffness

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Stiffening leads to lower vibration for the same value of damping

Design methodology : Influence of stiffness


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Design methodology : Influence of damping

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Design methodology : Influence of damping


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Remedial measures


What can you do if levels of vibrations are excessive ?

Remedial measure 1 : Increased damping

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Damping

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Types and origin of damping

Damping = dissipation of energy


Local damping models


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Material damping

Viscous damping

Loss factor – Hysteretic damping

Local damping models

Loss factor can be different for each material and frequency dependent (frequency domain computations)

In each material(time domain computations)

Those damping coefficients can be identified experimentally on small material samples

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Typical values of damping

Contributions to damping

Damping coefficients are usually derived from practice or measured if the structure is built


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Measurement of damping

Logarithmic decrement method Half-power bandwidth

Estimation of in the time domain Estimation of in the frequency domain

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Logarithmic decrement method

Free response


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Half-power bandwidth

Remedial measure 2 :The tuned mass damper (TMD)

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Application example : the tuned mass damper

Equations of motion:

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Application example : the tuned mass damper

Undamped vibration absorber (b=0)

for

Harmonic excitation:


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Undamped tuned mass damper

The damping device is tuned to the eigenfrequency of the primary system-> Reduces vibrations in a narrow band around eigenfrequency-> Amplification outside of this narrow band

If you choose you can cancel the vibration of the primary system at its natural frequency

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Pendulum tuned mass damper

Inertial coupling of the two systems

small


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Harmonic excitation:

for

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•Tuning of the PTMD based on the length of the pendulum

•Effect of the mass mainly on the spreading of the peaks


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Damped tuned mass damper

-Reduction of vibration is lower around eigenfrequency with b increasing-Reduces the amplification outside of the narrow frequency band-Existence of P and Q : points where all curves cross

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Optimal design of tuned mass dampers

Optimum damping is given by

P and Q are at equal height for

[ Den Hartog, 1954]


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Steps to follow to design a TMD

- The maximum mass of the device is decided fixing

- Based on this value, the frequency of the TMD is tuned :

- Which allows to compute the stiffnes of the TMD

- And finally the optimal damping is computed

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Optimal tuning rules for base excitation

Goal : minimize


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Damped pendulum tuned mass damper

Analogy with TMD[ Deraemaeker 2018]

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Application of TMDs to continous systems modeled with the finite element method

Reduce main system to a one dof system using as reference point the point of attachment of the TMD (in the direction of its motion)


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Single mode approximation

Point load at the position of the TMD

Application of TMDs to continous systems modeled with the finite element method

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Reduction of a finite element model to a one dof system

• Apply tuning rules using equivalent mass and stiffness

• Does not work for base excitation !


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Tuned mass damper attached to continuous systems

Ground motion (acceleration)

Example : predicting the response of a 10 storey building to earthquake

Without TMD With TMD tuned to first mode


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Undamped tuned mass damper

The building is excited at its natural frequency (2.62 Hz)

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Tuned mass damper in action


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TMDs applied to pedestrian bridges

Case study 1 : pedestrian induced vibrations of a footbridge


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Example of tuned mass dampers in structures

Millenium bridge, London


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Tuned mass damper in action on a bridge

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Tuned mass damper in action on a bridge


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TMDs applied to high-rise buildings

Case study 2 : Vibrations of high-rise buildings


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John Hancock Tower (Boston-1976)

Two TMDs of 2700 kN (approx 5.2x5.2x1m steel blocks)


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City Corp Center (New York - 1977)

Tuned mass damper

-400 Tons block installed at the top(2% of effective mass of first mode)

- 279m high- Fundamental period = 6.5s

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City Corp Center (New York - 1977)


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Chiba Port Tower (Japan - 1986)

Tuned mass damper : 15 tonsCan slide in two directions

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Damped pendulum tuned mass damper

Dampers

Tai Pei (Taiwan)


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Pendulum tuned mass damper example – Taipei 101

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Pendulum tuned mass damper example – Taipei 101


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Pendulum tuned mass damper example

Pendulum motion during earthquake, May 12, 2008

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Case study 3 : Machinery induced vibrations in a building



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Remedial measure 4 :Direct vibration isolation

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Vibration transmission

Vibration isolation

Isolation factor:

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Force transmitted to the ground:


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Vibration isolation

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Vibration isolation : isolation domain

•Choose k such that the isolation domain corresponds to the frequency content of the excitation signal f(t)•This results usually in the use of soft springs

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Vibration isolation : influence of damping

High damping- Low amplification around resonance - Reduced isolation

Ideal solution : frequency dependent damping (high at low frequencies, low at high frequencies)Example: rubber, elastomers …

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Low damping- High amplification around resonance - Good isolation at high frequencies

Vibration isolation : examples

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Case study 4 : Vibrations caused by traffic


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Vibration damping in railway tracks


Addition of damping


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Vibration isolation of railway tracks

Vibration isolation systems

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Remedial measures for earthquakes

Case study 2 : Vibrations of high-rise buildings

• Source of excitation• Effects• Design methodology

• Low/high tuning• Damping

• Remedial measures :• TMDs• Inverse vibration

isolation



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Remedial measure 5 :Inverse vibration isolation

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Inverse vibration isolation

The transmissibility is equal to the isolation factor previously definedThe aim is to ‘decouple’ the motion of the building from the ground motion

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Transmissibility :


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To isolate in the low frequency domain, we need k small, m high

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Inverse vibration isolation

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Inverse vibration isolation : examples


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Inverse vibration isolation : examples

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Inverse vibration isolation : precision microscope


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Inverse vibration isolation : domonstrators

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Inverse vibration isolation : real-life examples


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Vibration isolation versus damping


Isolation or damping ?


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Vibration damping of 5 storey building under ground excitation

Initial damping Added damping

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Vibration isolation of 5 storey building under ground excitation

• Building considered as a rigid body• Cut-off frequency to compute kt


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Cut-off frequency of 4 Hz

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f=4Hz

f=1Hz


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• Isolation and damping are very different mechanisms• Above cut-off frequency, if well designed, isolation is generally

more efficient (but might be more expensive) than damping.

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Dynamics of structures - homepages.ulb.ac.behomepages.ulb.ac.be/~aderaema/dynamics/6_DOS_Remedial_measu… · Dynamics of Structures 2019-2020 6. Remedial measures 14 27 Steps to