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Department of Civil Engineering Department of Civil Engineering IIT IIT Guwahati Guwahati

CE 532 Lecture 20-21 Vibration Absorber

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Page 1: CE 532 Lecture 20-21 Vibration Absorber

Department of Civil EngineeringDepartment of Civil Engineering

IIT IIT GuwahatiGuwahati

Page 2: CE 532 Lecture 20-21 Vibration Absorber

Vibration Absorber

9/21/20119/21/20119/21/20119/21/2011 CE 532 Lecture 20CE 532 Lecture 20CE 532 Lecture 20CE 532 Lecture 20----21: Vibration Absorber21: Vibration Absorber21: Vibration Absorber21: Vibration Absorber 2222

Auxiliary or

Absorbing

System

m2

k2z1

z2

Q=Q0sin(ωt)

Coupled System

Main Vibrating

System

m1

k1

k2z1 Coupled

Vibration

Absorbing

System

Page 3: CE 532 Lecture 20-21 Vibration Absorber

Vibration Absorber

� A 2DOF mass-spring system wherein an auxiliary or absorbing

system is mounted on the main system

� Principal of undamped vibration absorber

9/21/20119/21/20119/21/20119/21/2011 CE 532 Lecture 20CE 532 Lecture 20CE 532 Lecture 20CE 532 Lecture 20----21: Vibration Absorber21: Vibration Absorber21: Vibration Absorber21: Vibration Absorber 3333

� A system may vibrate excessively under a steady oscillatory force,

especially at the conditions of resonance

� Such vibrations are reduced or completely eliminated by coupling the

main vibrating system with an auxiliary system

� Vibration is finally transmitted to the auxiliary system, bringing the

main system to rest

Page 4: CE 532 Lecture 20-21 Vibration Absorber

Vibration Absorber

� When only main system exists

� Natural frequency of the main system

� Operating forcing function

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1 1 1m m k mω ω= =

( )0 sinQ Q tω=

� Static deflection of the main system under external force

� Vibration absorber is to be designed for the worst case i.e. for

the Resonance condition

� If such a design reduces or completely eliminates the vibration of the

main system, inadvertently at other frequencies of operation, the

vibration are automatically reduced, if not completely eliminated

� Resonance condition

0 1stz Q k=

mω ω=

Page 5: CE 532 Lecture 20-21 Vibration Absorber

Vibration Absorber

� The auxiliary or absorbing system is mounted on the main

system

� Natural frequency of the auxiliary system

� The mass ratio of the coupled system

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2 2 2a m k mω ω= =

2m

mµ η= =� The mass ratio of the coupled system

� The mounting of the auxiliary system creates a 2DOF system

which has its own natural frequencies of vibration and different

from that of the auxiliary system and the main system.

1m

mµ η= =

Page 6: CE 532 Lecture 20-21 Vibration Absorber

Vibration Absorber

� Amplitude of the main system (A1) and absorbing system (A2)

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20 0

21 1

1 22 2 2 22 2 2 2

1

,a

Q Q

k kA A

k k k k

ω

ω

ω ω ω ω

= =

− + − − − + − −

� Prime objective of the vibration absorber

� Bring the main system to rest – Eliminate any vibration in the main

system

2 2 2 22 2 2 2

2 2 2 21 1 1 1

1 1 1 1

a m a m

k k k k

k k k k

ω ω ω ω

ω ω ω ω

− + − − − + − −

1 0A =

Page 7: CE 532 Lecture 20-21 Vibration Absorber

Vibration Absorber

� In order to bring the main system to rest, the natural frequency

of the vibration absorber should be equal to the operating

frequency

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1If 0, aA ω ω= =

� Substituting the above condition in A2

� The force generated in the vibration absorber is equal and opposite to

that of the operating force

� No net force is transmitted to the main system, and the main system is

brought to rest

( )2 2 1

2 2 1 2 2 00 1 2

stst

A A kA k k z A k Q

Q k z k= = − ⇒ = − ⇒ = −

Page 8: CE 532 Lecture 20-21 Vibration Absorber

Vibration Absorber

� Efficiency of a vibration absorber is best appreciated when the

main system vibrates in resonance or near resonance

� Under such condition, the absorber is designed in such a way

that the main system is brought to absolute or near rest

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mω ω=

that the main system is brought to absolute or near rest

condition

� Best performance from the absorber is obtained at the operating

frequency corresponding to the natural frequency of the main system

alone

aω ω=

2 1 2 2

2 1 1 1a m m

k k k m

m m k mω ω µ= ⇒ = ⇒ = =

Tuned Undamped

Vibration absorber

Page 9: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

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m2

k2z1

z2

Q=Q0sin(ωt)

m1

k1

k2z1

Page 10: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

� The Tuned Undamped Vibration Absorber is a 2DOF system,

which has its own natural frequencies of vibration ωn (ωn1 and

ωn2)

� Under the best and the most efficient performance of the

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� Under the best and the most efficient performance of the

vibration absorber, the main system would suffer zero

displacement at the operating frequency equal to the natural

frequency of the main system

� However, this coupled system would move into resonance as a

whole at some other natural frequency ω* which is equal to

natural frequencies of the coupled system ωn (ωn1 and ωn2)

Page 11: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

� The amplitudes of vibration of the main system and the

absorber system

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20 0

21 1

1

,a

Q Q

k kA A

ω

ω

= =

� At ωa=ωm=ω, A1=0 i.e. the main system has zero vibration is completely

brought to rest, while the absorbing system undergoes vibration with an

amplitude equal to the inverse of mass-ratio and in an opposite direction

to the excitation

1 11 22 2 2 2

2 2 2 22 2 2 2

1 1 1 1

,

1 1 1 1

a

a a a a

kA A

k k k k

k k k k

ω

ω ω ω ω

ω ω ω ω

= =

− + − − − + − −

11 2 2

2

10,

m

kA A A

k µ= = − ⇒ =

Page 12: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

� Let the whole Tuned Absorber system moves into resonance at

some operating frequency ω*=ωn

� Amplitudes of vibration of main system and the absorber

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20 0

21 nQ Qω

� Under conditions of resonance

0 02

1 11 22 2 2 2

2 2 2 22 2 2 2

1 1 1 1

1

,

1 1 1 1

n

a

n n n n

a a a a

Q

k kA A

k k k k

k k k k

ω

ω ω ω ω

ω ω ω ω

− = =

− + − − − + − −

1 2

2 22 2

2 21 1

,

1 1 0n n

a a

A A

k k

k k

ω ω

ω ω

= ∞ = ∞

⇒ − + − − =

Page 13: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

� The two natural frequencies of

the coupled absorber system

are

9/21/20119/21/20119/21/20119/21/2011 CE 532 Lecture 20CE 532 Lecture 20CE 532 Lecture 20CE 532 Lecture 20----21: Vibration Absorber21: Vibration Absorber21: Vibration Absorber21: Vibration Absorber 13131313

1 2

2 22 2

2 21 1

2 2

2 2

,

1 1 0

1 1 0

n n

a a

n nm m

A A

k k

k k

ω ω

ω ω

ω ωµ µ

ω ω

= ∞ = ∞

⇒ − + − − =

⇒ − + − − =

2

1 m mµ µω ω µ

= + − +

( )

( ) ( )

2 2

22 2

2 2

22

2

2 2

2

1 1 0

2 1 0

2 2 4

2

12 4

m m

a a

n nm

a a

m mn

a

n m mm

a

µ µω ω

ω ωµ

ω ω

µ µω

ω

ω µ µµ

ω

⇒ − + − − =

⇒ − + + =

+ ± + −⇒ =

⇒ = + ± +

1

2

2

12 4

12 4

m mn a m

m mn a m

µ µω ω µ

µ µω ω µ

= + − +

= + + +

Page 14: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

� Relationship between mass ratio and natural frequencies of the

coupled absorber system

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Page 15: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

� Inferences from the relationship between mass ratio and the

natural frequencies

� The curve at the point ω/ωa=1 at µm=0 represents the condition

� The mass of the absorbing system is negligible to that of the main system.

� The stiffness of the absorbing system is negligible to that of the main system

� This implies that the absorbing system in this case is made up of a very small mass

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� This implies that the absorbing system in this case is made up of a very small mass

resting on a very flexible spring

� Under such condition

� The vibration of the main system is brought to absolute rest

� The absorbing system vibrates with infinite amplitude

� For any other mass ratio (µm>0 )

� The main system is brought to absolute rest at ω/ωa=1

� For ω/ωa=1, the absorber system vibrates with an amplitude inversely

proportional l to µm

Page 16: CE 532 Lecture 20-21 Vibration Absorber

Tuned Undamped Vibration Absorber

� Inferences from the relationship between mass ratio and the

natural frequencies

� For each µm, the two natural frequencies of the coupled 2DOF system

are obtained

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are obtained

� Higher the µm, larger is the distance between the two natural frequencies of

the coupled system

� The rate of increase of the second principal frequency is more than the

fundamental frequency

� Fundamental frequency of the coupled system governs the choice of the

absorbing system to be used

Page 17: CE 532 Lecture 20-21 Vibration Absorber

Frequency Response Curve

� The amplitudes of vibration of the main system and the

absorber system

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2

21 2

11

,aA A

ω

ω

= =

� Plot of the above normalized amplitudes with respect to the normalized

frequency provides the frequency response curve for the couples

vibration absorber system

1 2

2 2 2 2

2 2 2 2

1,

1 1 1 1

a

st stm m m m

a a a a

A A

z zω ω ω ωµ µ µ µ

ω ω ω ω

= =

− + − − − + − −

Page 18: CE 532 Lecture 20-21 Vibration Absorber

Frequency Response Curves

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Frequency response curve of the main system

Page 19: CE 532 Lecture 20-21 Vibration Absorber

Frequency Response Curves

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Frequency response curve of the absorber system

Page 20: CE 532 Lecture 20-21 Vibration Absorber

Frequency Response Curves

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Comparison of Frequency response curves of the main and absorber system

Page 21: CE 532 Lecture 20-21 Vibration Absorber

Frequency Response Curves

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Comparison of Frequency response curves of the main and absorber system

Page 22: CE 532 Lecture 20-21 Vibration Absorber

Frequency Response Curves

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Comparison of Frequency response curves of the main and absorber system

Page 23: CE 532 Lecture 20-21 Vibration Absorber

Performance of a Vibration Absorber

� The best performance of the vibration absorber is obtained

when the absorber so designed has its natural frequency equal

to the operating resonance frequency of the system, which, in

turn, is equal to the natural frequency of the main system

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� As long as the operating frequency remains constant and nearly

equal to the natural frequency of the absorber, the absorbing

system will render best and the most efficient performance i.e.

the main system will be provided absolute rest

Page 24: CE 532 Lecture 20-21 Vibration Absorber

Performance of a Vibration Absorber

� If the operating frequency changes due to some reason after the

design and installation of the absorbing system, then the

efficiency of the absorbing system will be apparently reduced

in the sense that the objective of providing absolute rest to the

main system will no longer be fulfilled

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main system will no longer be fulfilled

� If the shift in the operating frequency is towards or near to nay

one of the natural frequencies of the coupled system, the whole

system will experience resonance, and the entire system will

vibrate with maximum amplitude

Page 25: CE 532 Lecture 20-21 Vibration Absorber

Steps to Design Vibration Absorber

1. Properties of the main system are known (m1, k1, ωm)

2. Worst case to be considered for the design of vibration

absorber � When the main system moves to resonance

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� Hence, the operating frequency is known and equal to the natural

frequency of the main system (ω=ωm)

3. Decide by how much the nearest natural frequency of the

coupled system should be pushed away from the resonant

frequency of the main system

� Say, the difference is ‘x’%1 1

100n m

xω ω

= −

Page 26: CE 532 Lecture 20-21 Vibration Absorber

Steps to Design Vibration Absorber

4. Use ωm to find the mass-ratio for the absorber system to be

designed

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2

1 1 ,2 4

m mn a m a m

µ µω ω µ ω ω

= + − + =

5. Find the properties of the absorber from the mass-ratio

2 2

1 1m

m k

m kµ = =

22

1

1

1n

a

mn

a

ω

ωµ

ω

ω

− ⇒ =

Page 27: CE 532 Lecture 20-21 Vibration Absorber

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Thank You for Patient Hearing