vibration absorbers/neutralisers · 2015. 2. 13. · vibration absorber/tuned mass damper ....
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Vibration Absorbers/Neutralisers
Professor Mike Brennan
Vibration absorbers/neutralisers
Vibration absorber/neutraliser
• What does it do?
• Absorber or neutraliser
• Mobility approach – effectiveness of the device
• Applications
m
k c
Vibration absorbers/neutralisers
j tf Fe
j tx Xe
structure
The Vibration Absorber – What does it do?
m
k c
frequency
X
F
Tunable vibration absorbers - some terminology
X
F
frequency
Natural frequency
• Absorber: Tuned to suppress the
response at a
troublesome resonance
frequency
frequency
X
F
Forcing frequency
• Neutraliser: Tuned to suppress the
response at a
troublesome forcing
frequency
General framework
Vibrating system 1
3
2
1F
2F
Question:
What effect does the auxiliary system
have on the velocity of the vibrating
system at points (2) and (3)?
A
AF
Auxiliary
system
m
c
k
1 11 12 13 1
2 21 22 23 2
3 31 32 33 3
V Y Y Y F
V Y Y Y F
V Y Y Y F
The equations describing the system are
A A AV Y F
General framework
Vibrating system 1
3
2
1F
2F
The boundary conditions are
A
AF
Auxiliary
system
3 21 3231
1 22 A
V Y YY
F Y Y
which results in 2 21
221 1A
V Y
YF
Y
3 (no force appl0 ied)F
2 (force equilibrium)AF F
2 (continuity of motion) AV V
Notes
• Even if YA = 0, i.e., the impedance is infinite, then V3 ≠ 0.
• If YA → 0, then V2 → 0.
A Auxiliary
system
Place auxiliary system at source
Vibrating system 1
3 1F
Notes
• The whole system can only be brought to rest if the auxiliary system is
fitted at the point of excitation.
• If the auxiliary system is fitted at a remote point the only this point can
be brought to rest
2 1 point ≡ point
Replace which results in
3 31
111 1A
V Y
YF
Y
21 11 with Y Y
22 11 with Y Y
32 31 with Y Y
• Thus if YA → 0, then V3 → 0.
Vibration absorber/Tuned mass damper
Vibration absorber/Tuned mass damper
m
k
j tFe
j t
sX e
• Tuned to a troublesome resonance of a structure
ma
ka ca
absorber
mode of
structure
aZ sZ
j t
sV e
j tFe
structure absorber
1s
a s
V
F Z Z
• Mobility of structure with absorber attached is
Vibration absorber/Tuned mass damper
aZ sZ
j t
sV e
j tFe
structure absorber
1s
a s
V
F Z Z
s
kZ j m
j
• The impedance of the
structure is given by
• The impedance of the
absorber is given by
1
1 1a
a aa
Z
k j mc
j
Tuned vibration absorber
X
F
frequency
Typical response of an undsamped 2DOF
system
m
k
F sX
ma
ka
0 0.5 1 1.5 2 2.5-6
-5
-4
-3
-2
-1
0
1
2
3
4
aX
frequency ratio
dis
pla
ce
me
nt ra
tio
0.3am
m
n
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
2
4
6
8
10
12
14
16
18
20
frequency ratio n
am
plit
ud
e r
atio
• Mass ratio
Vibration absorber - notes
• Important parameters are mass ratio and damping in the absorber am
m
• Tuned condition for an undamped absorber is a
a
k k
m m
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.80.6
0.7
0.8
0.9
1
1.1
1.2
1.3
1.4
1.5
1.6
r
n
mass ratio
0.05
0.2
Frequency ratio
X
F
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
2
4
6
8
10
12
14
16
18
20
n
ma
ka
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
2
4
6
8
10
12
14
16
18
20
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
2
4
6
8
10
12
14
16
18
20
The vibration absorber effect of damping
m
k c
F X
ca
Original
structure
Absorber
attached
with damping
Absorber
attached
no damping
0.6 0.7 0.8 0.9 1 1.1 1.2 1.3 1.40
2
4
6
8
10
12
14
16
18
20
The absorber – some key parameters
X
F
0
opt
Frequency ratio n
opt 3
3
8 1
•The optimum damping is
given by
i.e., always lower than the
troublesome resonance
frequency
•To ensure the amplitudes at
points A and B are the same,
the absorber should be tuned to
1
1a
n
A B
The absorber – example
m
k c
F X
ma
ka ca
Mass ratio 0.1am
mOptimum Damping
opt 3
30.17
8 1
frequency
X
F
(a) Tension-compression
Types of absorber/tuned mass damper
(b) Shear
(c) Bending
Some Applications for the Absorber
Some Applications for the Absorber
Some Applications for the Absorber
Some Applications for the Absorber
Some Applications for the Absorber
The Millenium Bridge in London
Some applications for the absorber
The millenium bridge in London
Some applications for the absorber
The Millenium Bridge in London
Some applications for the absorber
The Millenium Bridge in London
•52 vibration absorbers
•2 Tonnes each
Stockbridge Damper
Different Manifestations of
vibration absorbers
Tuned Liquid Column Absorber
H
B
2
n
g
B H
Shunted Piezoelectric Absorber
1 n
LC
The Smart Ski (ACX.com)
The Smart Ski (ACX.com)
Piezo patches
The Smart Bat (ACX.com)
Second bending mode (670 Hz)
Third bending mode (1252 Hz)
Fundamental bending mode (215 Hz)
The Smart Bat (ACX.com)
Tuned vibration neutralisers
frequency
X
F
Forcing frequency
• Neutraliser: Tuned to suppress the response at a troublesome
forcing frequency
Tuned vibration neutralisers
nZ sZ
j t
sV e
j tFe
structure neutraliser
fitted
free
1
1
s
ns
s
V
ZV
Z
• Divide (1) by (2) gives
fitted 1s
n s
V
F Z Z
• Vibration of structure with
neutraliser fitted
(1)
free 1s
s
V
F Z
• Vibration of structure alone
(2)
(fitted) (free)Thus if then n s s sZ Z V V
Impedance of a neutraliser
kn
mn
V cn
F
total
1
1 1
m k c
Z
Z Z Z
total 2
1 2
1 2
n n
n n
j m jZ
j
natural frequency of neutraliser
n
• If the neutraliser is tuned
such that
(max)2n n
n
mZ
and assuming that gives 1
The the maximum impedance
of a neutraliser
• increases with frequency
• increases with mass
• decreases with damping
• is real and hence damping-like
frequency
X
F
Forcing frequency
Simple example – mass-like structure
Mass-like behaviour of host structure
mn
kn cn
m F
sV
n
( fitted)
( free)
s
s
V
V
frequency
010
, n nn
n
k m
m m
Simple example – mass-like neutraliser
sZ j m
• Impedance of structure
given by
fitted
free tunedtuned
2s s
ns
V Z
V Z
• Assuming that
then (max)n sZ Z
2
Motion of neutraliser mass (when tuned)
( fitted)
Velocity of neutraliser mass
Velocity of structuren
s
VT
V
mn
kn cn
m FsV
nVwhere
2
1 2
1 2
n
n n
jT
j
When the device is tuned, i.e., n
( fitted) tuned
1
2n
s
V
V (1)
fitted
freetuned
2s
s
V
V
Now (2)
Dividing (1) by (2) gives free
tuned
1n
s
V
V
i.e., the mass ratio controls the
motion of the neutraliser mass
at its tuned frquency
mn
kn cn
m F
sV
n
( fitted)
( free)
s
s
V
V
frequency
010
nn
n
k
m
Bandwidth of a neutraliser
2 1 2n
B
• The bandwidth is defined as
• This means that a small ζ gives
a large attenuation, but a small
bandwidth 1 2
3 dB
Application of neutraliser - Boeing CH - 47C
Three neutralisers are
installed and tuned to
the blade passage
frequency of
approximately 11 Hz
Summary
• Vibration absorbers/tuned mass dampers – Used to suppress vibration at troublesome resonance
frequencies
• Vibration neutralisers – Used to suppress vibration at a forcing frequency
• Optimum parameters
• Applications
References
• C.M. Harris, 1987, Shock and Vibration Handbook, Third
Edition, McGraw Hill.
• R.G. White and J.G. Walker, 1982, Noise and Vibration, Ellis
Horwood Publishers.
• J.P. Den Hartog, 1956. Mechanical Vibrations, Second Edition.
Dover Publications
• S.S. Rao, 2003, Mechanical Vibrations, Fourth Edition, Prentice
Hall.