Tuned Mass Dampersa mass that is connected to a structureby a spring and a damping elementwithout any other support,in order toreduce vibration of the structure.

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 oneof their mode shapes by wind,

Taipeh 101

stairs, spectator stands, pedestrian bridges excited by marching orjumping people. These vibrations are usually not dangerous for thestructure itself, but may become very unpleasant for the people,

steel structures like factory floors excited in one of their naturalfrequencies by machines , such as screens, centrifuges, fans etc.,

ships exited in one of their natural frequencies by the main enginesor even by ship motion.

SDOF System

)tsin()2()1(

1kp

u222

1

01

1

11 m

keigenfrequency:

damping ratio of Lehr:11

1

m2c

tp cos0

1

11 m

k

• 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

• Strengthen the structure to get a higher

eigenfrequency

mEI

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

•Application of dampers

•Application of tuned mass dampers

2 DOF System

tcosp0

tCtCu

tCtCu

sincos

sincos

432

211

0uucuukum

tcospuuc

uukukum

12212222

0212

2121111

solution:

differential equations:

22

21max,1 CCu

24

23max,2 CCu

linear equation system by derivation of the solutionand application to the differential equations:

:= C [ ]p0 ( ) m1 2k1 k2 p0 c p0 k2 p0 c

1

0,1 k

pu stat

1

11 m

k

2

22 m

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:

1

statu

u

,1

max,1

10.0

32.0

0

All lines meet in the points S and T

S T

statstat u

u

u

u

,1

1

,1

1 )()0(

2

2

2

11

2

1 222222

,TS→

11

1

1 222222

22

→

→ 022

21

2222

24

:stat,1u1u

in

2T

2T

stat,1T,1

2S

2S

stat,1S,1

1

uu

1

uu

Optimisation of TMDfor the smallest deformation:

1

1Optimal ratio of eigenfrequencies: → Optimal

spring constant2k

3,1,1,118

3

statTS uuuMinimize

→ Optimal damping constant: 318

3

opt

→

TS uu ,1,1

122

T2

S

2

12

222

T2

S

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

• Different Assumptions of Youngs Modulus and Weights

• Increased Main Mass caused by the load

Adjustment:

•Large displacement of the damper mass

Plastic deformation of the spring Exceeding the limit of deformation

Problem:

Realization

damping of torsional oscillation

400 kg - 14 Hz

Millennium Bridge

Mass damper on an electricity cable

Pendular dampers