vibration absorbers/neutralisers · 2015. 2. 13. · vibration absorber/tuned mass damper ....
TRANSCRIPT
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Vibration Absorbers/Neutralisers
Professor Mike Brennan
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Vibration absorbers/neutralisers
Vibration absorber/neutraliser
• What does it do?
• Absorber or neutraliser
• Mobility approach – effectiveness of the device
• Applications
m
k c
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Vibration absorbers/neutralisers
j tf Fe
j tx Xe
structure
The Vibration Absorber – What does it do?
m
k c
frequency
X
F
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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
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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
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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.
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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.
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Vibration absorber/Tuned mass damper
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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
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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
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Tuned vibration absorber
X
F
frequency
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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
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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
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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
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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
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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
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(a) Tension-compression
Types of absorber/tuned mass damper
(b) Shear
(c) Bending
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Some Applications for the Absorber
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Some Applications for the Absorber
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Some Applications for the Absorber
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Some Applications for the Absorber
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Some Applications for the Absorber
The Millenium Bridge in London
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Some applications for the absorber
The millenium bridge in London
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Some applications for the absorber
The Millenium Bridge in London
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Some applications for the absorber
The Millenium Bridge in London
•52 vibration absorbers
•2 Tonnes each
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Stockbridge Damper
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Different Manifestations of
vibration absorbers
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Tuned Liquid Column Absorber
H
B
2
n
g
B H
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Shunted Piezoelectric Absorber
1 n
LC
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The Smart Ski (ACX.com)
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The Smart Ski (ACX.com)
Piezo patches
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The Smart Bat (ACX.com)
Second bending mode (670 Hz)
Third bending mode (1252 Hz)
Fundamental bending mode (215 Hz)
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The Smart Bat (ACX.com)
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Tuned vibration neutralisers
frequency
X
F
Forcing frequency
• Neutraliser: Tuned to suppress the response at a troublesome
forcing frequency
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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
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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
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frequency
X
F
Forcing frequency
Simple example – mass-like structure
Mass-like behaviour of host structure
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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
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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
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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
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Application of neutraliser - Boeing CH - 47C
Three neutralisers are
installed and tuned to
the blade passage
frequency of
approximately 11 Hz
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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
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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.