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Tuned Mass Dampers

a mass that is connected to a structureby a spring and a damping element

without any other support,in order to

reduce vibration of the structure.

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Tuned mass dampers are mainly used in the following applications:

tall and slender free-standing structures (bridges, pylons of bridges,

chimneys, TV towers) which tend to be excited dangerously in one

of their mode shapes by wind,

Taipeh 101

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stairs, spectator stands, pedestrian bridges excited by marching or

jumping people. These vibrations are usually not dangerous for the

structure itself, but may become very unpleasant for the people,

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steel structures like factory floors excited in one of their natural

frequencies by machines , such as screens, centrifuges, fans etc.,

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ships exited in one of their natural frequencies by the main engines

or even by ship motion.

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SDOF System

)tsin()2()1(

1

k

pu

2221

01

1

11 m

keigenfrequency:

damping ratio of Lehr:11

1

m2

c

tp cos0

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1

11 m

k

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Thin structures with low damping have a high

peak in their amplification if the frequency of

excitation is similar to eigenfrequency

High dynamic forces and deformations

Solutions:

Strengthen the structure to get a higher

eigenfrequency

Application of dampers

Application of tuned mass dampers

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Strengthen the structure to get a higher

eigenfrequency

m

EI

L2f

2

2

1

Eigenfrequency of a beam:

Doubling the stiffness only leads to multiplicationof the eigenfrequency by about 1.4.

Most dangerous eigenfrequencies forhuman excitation: 1.8 - 2.4 Hz

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Application of dampers

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Application of tuned mass dampers

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2 DOF System

tcosp0

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tCtCu

tCtCu

sincos

sincos

432

211

0uucuukum

tcospuucuukukum

12212222

0212

2121111

solution:

differentialequations:

22

21max,1 CCu

24

23max,2 CCu

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linear equation system by derivation of the solution

and application to the differential equations:

:=C [ ]p0 ( ) m12

k1 k2 p0 c p0 k2 p0 c

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1

0

,1k

pu stat

1

11m

k

2

22m

k

1

2

1

22

2

2

m

c

1

2

m

m

static deformation:

eigenfrequencies:

Damping ratio of Lehr:

mass ratio:

22222222222

22222

,1

max,1

114

4

statu

u

ratio of frequencies:

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statu

u

,1

max,1

10.0

32.0

0

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All lines meet in the points S and T

S T

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statstat u

u

u

u

,1

1

,1

1 )()0(

22

211

21

22

222

2,TS

111

1 222222

22

02

2

2

12

22224

:stat,1u

1uin

2

T

2

T

stat,1T,1

2S2S

stat,1S,1

1

uu

1

uu

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Optimisation of TMD

for the smallest deformation:

11Optimal ratio of

eigenfrequencies: Optimal

spring constant2

k

3,1,1,1 183

statTS uuuMinimize

Optimal damping constant:

3

18

3

opt

TS uu ,1,1

122

T

2

S

2

12

222

T

2

S

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Ratio of masses:

The higher the mass of the TMD is, the better is the damping.Useful: from 0.02 (low effect) up to 0.1 (often constructive limit)

Ratio of frequencies:

0.98 - 0.86

Damping Ratio of Lehr:

0.08 - 0.20

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Different Assumptions of

Youngs Modulus and

Weights

Increased Main Masscaused by the load

Adjustment:

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Large displacement of the damper massPlastic deformation of the spring

Exceeding the limit of deformation

Problem:

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Realization

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damping of torsional oscillation

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400 kg - 14 Hz

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Millennium Bridge

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Mass damper on an electricity cable

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Pendular dampers

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