Stroscher.ppt

Tuned Mass Dampers
a mass that is connected to a structure
by a spring and a damping element
without any other support,in order to
reduce 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 one
of their mode shapes by wind,
Taipeh 101

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,

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

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

SDOF System
)
t
sin(
)
2
(
)
1
(
1
k
p
u
2
2
2
1
0
1 













1
1
1 m
k


eigenfrequency:
damping ratio of Lehr:
1
1
1
m
2
c





 
t
p 
 
cos
0

• 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
m
EI
L
2
f 2
2
1




Eigenfrequency of a beam:
Doubling the stiffness only leads to multiplication
of the eigenfrequency by about 1.4.
Most dangerous eigenfrequencies for
human excitation: 1.8 - 2.4 Hz

•Application of tuned mass dampers

2 DOF System
 
t
cos
p0 



   
   
t
C
t
C
u
t
C
t
C
u
















sin
cos
sin
cos
4
3
2
2
1
1
 
   
    0
u
u
c
u
u
k
u
m
t
cos
p
u
u
c
u
u
k
u
k
u
m
1
2
2
1
2
2
2
2
0
2
1
2
2
1
2
1
1
1
1





























solution:
differential equations:
2
2
2
1
max
,
1 C
C
u 

2
4
2
3
max
,
2 C
C
u 


linear equation system by derivation of the solution
and application to the differential equations:
:=
C [ ]
p0 ( )
 
m1 
2
k1 k2 p0 c  p0 k2 p0 c 

1
0
,
1
k
p
u stat 
1
1
1
m
k


2
2
2
m
k


1
2


 
1


 
2
2
2
2 




m
c
1
2
m
m


static deformation:
eigenfrequencies:
Damping ratio of Lehr:
mass ratio:
 
     
 2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
2
,
1
max
,
1
1
1
4
4































stat
u
u
ratio of frequencies:

1


 
stat
u
u
,
1
max
,
1


 10
.
0


32
.
0


0



All lines meet in the points S and T
S
 T


stat
stat u
u
u
u
,
1
1
,
1
1 )
(
)
0
( 


 

 































2
2
2
1
1
2
1 2
2
2
2
2
2
,T
S
→
     1
1
1
1 2
2
2
2
2
2
2
2
















→
→ 0
2
2
2
1
2
2
2
2
2
4














 :
stat
,
1
u
1
u
in 



2
T
2
T
stat
,
1
T
,
1
2
S
2
S
stat
,
1
S
,
1
1
u
u
1
u
u















Optimisation of TMD
for the smallest deformation:




1
1
Optimal ratio of
eigenfrequencies: → Optimal
spring constant
2
k
 3
,
1
,
1
,
1
1
8
3







 stat
T
S u
u
u
Minimize
→ Optimal damping constant:
 3
1
8
3







opt
→
T
S u
u ,
1
,
1 






1
2
2
T
2
S










2
1
2
2
2
2
T
2
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:

damping of torsional oscillation

Mass damper on an electricity cable

Stroscher.ppt

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