vibration absorbers/neutralisers · 2015. 2. 13. · vibration absorber/tuned mass damper ....

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.