Tuned Mass Dampers
description
Transcript of Tuned Mass Dampers
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