-
Principle of tuned-mass vibration absorber:
Problem: Periodic excitation of vibrating system (resonant or
not)
Forced vibration can be cancelled by adding an auxiliary mass
tunedsuch that
11 mk=
-
210101
11
011
1111
011
)cos(0
)cos(
0)()cos()(
==
==
=
=+=++
mFkFA
tAxx
tFxk
xxkxmtFxxkkxxm
&&
&&
For auxiliary mass vibrates in such a manner that the force
exerted on m compensates the external force:
11 mk=
-
Problems/Limits of tuned-mass vibration absorbers:
Amplitude of auxiliary mass is limited by absorber design.Above
critical force amplitude absorber becomes saturated and absorber
action disappears.
If we try to remove resonant excitation (202=k/m): Adding
theauxiliary mass m1
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Case study 1: North Stand, Highbury Stadium (Arsenal),
1993-2006
1992-93 Demolition of old stand and replacement by new
all-seaterstand using lightweight construction methods.
natural frequency of vertical vibration mode of upper tier of
the fullyoccupied stand slightly above 1 Hz
-
Problem: Resonant excitation of vertical mode by periodic
vertical motionsof the Arsenal supporters.
Solution: Attachment of tunable vibration absorbers to the upper
tier
-
( ) )( elongation spring ntdisplaceme mass :relations Kinematic
1BCABABA BC1 xxlllxxlx
==
+=
( )( )2BCAB1eff1eff11
1BCABAB1BC11
1
,0)(0)(
: massfor motion ofequation :moments of Balance
llkkxxkxmxxlllklxm
m
==+=+
&&
&&
k1
-
( )
BC
1eff
01eff
2BCABeff1eff11
changingby tunedbecan constant spring effective , if cancelled
is stand of vibration
)cos()(,0)(
:mass)absorber (stand system for total system of Equations
lmk
tFkxxxkxmllkkxxkxm
=
=++==+
+
&&
&&
-
Solution of vibration problem by frustrating anthropogenic
excitation:absorber is automatically tuned to dominant vibration
frequency.
In spite of ~ 5000 supporters on stand: only small auxiliary
mass is needed(~ 2% of mass of the structure) since lack of
feedback prevents the supportersfrom synchronising their
motions.
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Case study 2: The worlds largest tuned mass vibration
absorber:
http://www.taipei-101.com.tw/en/OB/about/damper.asp
Taipei 101:
Until 2009 the worlds tallest building Situated near a major
earthquake fault Situated in a Typhoon area
Use of a tunable pendulum to control vibration of the building
under extremewind loads.
Pendulum mass: 730 tonsPendulum length (tunable) around 12m
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How does it work?
lxx
mgmgmgl
xx
=
1
1
tan
sin
0)(
0)cos()(
11
01
=+
=+
xxl
mgxm
tFxxl
mgKxxM
&&
&&
-
lg
km
lmgk
xxkxmtFxxkKxxM
xxl
mgxm
tFxxl
mgKxxM
==
=
=+=++
=+
=+
1
1
111
011
11
01
0)()cos()(
0)(
0)cos()(
&&
&&
&&
&&
These equations are equivalent to those for a vibration
absorber:
with spring stiffness
The forced vibration of the main mass is eliminated if
The absorber can be tuned to changing frequenciesby adjusting
the free length of the pendulum
-
In Taipei 101: pendulum natural frequency (about 1/7 Hz)
corresponds tofrequency of the fundamental flexural mode of the
tower structure
-> control of resonant excitation by gusting winds
