Professor Mike Brennan - UNESP · Forced vibration – effect of damping Log frequency Log 1 k k m x c FejtZ Frequency Regions 2 2( ) 2 1 F k m cZ X F X ... Assume that the loss factor

Vibration Damping

Professor Mike Brennan

Vibration Control by Damping

• Review of damping mechanisms and effects of

damping

– Viscous

– Hysteretic

– Coulomb

• Modelling and characteristics of viscoelastic damping

• Application of constrained and unconstrained layer

damping

• Measurement of damping

Damping

• material damping (internal friction, mechanical hysteresis)

• friction at joints

• added layers of materials with high loss factors (e.g. visco-

elastic materials)

• hydraulic dampers (shock absorbers, hydromounts)

• air/oil pumping: squeeze-film damping

• impacts (conversion of energy into higher frequency vibration

which is then dissipated)

• acoustic radiation - vibration energy converted into sound (only

important for lightly damped structures or heavy fluid loading,

e.g. light plastic structures)

• structural ‘radiation’ - transport of vibration into adjacent

structures or fluids

dissipation of mechanical energy (usually conversion into heat)

Review of Damping Mechanisms

Viscous damping

k

m x

c

The equation of motion is

0cx kxm x

inertia

force stiffness

force

damping

force

Example - automotive suspension system

Free vibration effect of damping

The underdamped displacement of the mass is given by

sinnt

dx Xe t

= Damping ratio = 2 0 1nc m

n = Undamped natural frequency = k m

d = Damped natural frequency = 21n

= Phase angle

Exponential decay term Oscillatory term

ntx Xe

Free vibration effect of damping

time

2d

d

T

d

sinnt

dx Xe t

Damping ratio

Damping perioddT

Phase angle

Hysteretic Damping

2

1X

F k m j c

k

m x

c

j tFe

m x

j tFe

1k j

Loss factor

Equation of motion

1 j tmx k j x Fe j tmx cx kx Fe

2

1X

F k m jk

Assuming a harmonic response leads to j tx Xe

Setting responses to be equal at resonance gives

Hysteretic damping force is in-phase with velocity and is proportional to displacement

Viscous damping force is in-phase with and proportional to velocity

2 where 2c mk

Hysteretic Damping

k

m x

c

f

m x

f

1k j

Hysteretic damping model with a constant stiffness and loss factor

is only applicable for harmonic excitation

( )f t

t

t

( )x t

( )f t

t

causal acausal

t

( )x t

Coulomb Damping

Coulomb damping is caused by sliding or dry friction

k

m

x

f

Fd = friction force (always

opposes motion)

+ve

-ve

d

d

x F

x F

• The damping force is independent of velocity once the motion is initiated,

but the sign of the damping force is always opposite to that of the velocity.

• No simple expression exists for this type of damping. We can equate the

energy dissipated per cycle to that of a viscous damper

dF

x

0

0 0

0

d n

n

F F x

x

F x

is the reaction forcenF mg

How to compare damping mechanisms?

Energy dissipated by a viscous damper

x

c dF

dF cx

Energy lost per cycle is

work done = force distanceE

0

T

d d

dxE F dx F dt

dt

Now if

sinx X t

then cosx X t

2

2

0

E cx dt

(1)

2

2 2 2

0

cosE c X t dt

Substitute into (1) gives

which evaluates to give

2 E c X

Returning to Coulomb Damping

k

m

x

f

• Since the damping force Fd = Fn is constant and the distance travelled

in one cycle is 4X, the energy dissipation per cycle is:

4 nE F X

Noting that 2

eqE c X

Gives an equivalent viscous damping coefficient 4 n

eq

Fc

X

Note that this is a non-linear form of damping – dependent upon amplitude

Returning to Coulomb Damping

log frequency

logV

F

F 4F V

• Illustration of the non-linear effects of Coulomb damping

Coulomb Damping

• Illustration of the non-linear effects of Coulomb damping

time

Equivalent viscous damping forces

and coefficients

Damping

Mechanism

Damping Force Equivalent

Viscous

Damping

Viscous

Hysteretic

Coulomb

cx

k x

nF

c

k

4 nF

X

Forced vibration – effect of damping

Log frequency

Log

1

k

k

m x

c

j tFe

Frequency Regions

222 )()(

1

cmkF

X

F

X

Low frequency 0 Stiffness controlled

Stiffness

controlled

kFX /1/

Resonance 2 k m Damping controlled

Damping

controlled

cFX /1/

High frequency 2

n Mass controlled

Mass

controlled

2/1/ mFX

Example: point mobility of 1.5 mm steel plate 0.6 0.5 m

= 0.01

= 0.1

infinite plate

10 1

10 2

10 3

10 4

10 -5

10 -4

10 -3

10 -2

10 -1

10 0

Frequency (Hz)

Mo

bili

ty (

m/N

s)

• Damping is only effective at controlling the response around the

resonance frequencies.

• At high frequencies the mobility tends to that of an infinite structure

and damping has no further effect.

Visco-elastic materials

• Visco-elastic materials (plastics, polymers, rubbers etc) have non-

linear material laws.

• For a harmonic input, visco-elastic materials are defined in terms of

their complex Young’s modulus E(1+j) or shear modulus G(1+j).

• Alternatively, we can write E = ER + jEI where ER is called the

storage modulus and EI is called the loss modulus.

• The parameters E, G and are dependent on

– frequency

– temperature

– strain amplitude

– preload

• Frequency and temperature dependencies are equivalent (higher

frequency is equivalent to lower temperature).

A model with a constant modulus and constant loss factor is often a good approximation

in a limited frequency region. However, it gives non-causal response in the time domain

Typical material damping

Visco-elastic damping – material properties

Simple model

• Combination of Voigt model

Maxwell model

k c

k

c

c

2k

1k

j tFe

j tXe

Dynamic stiffness is

2

2 1 2

2

1 1

22

1

1

k k kj

c k k cF

X

c k

Returning to Complex Stiffness

x

F ( ) 1 ( )k j

( ) 1 ( )F

k jX

Stiffness determined from ( ) ReF

kX

Loss factor determined from

Im

( )

Re

F

X

F

X


Stiffness

2

2 1 2

2

1 1

22

1

( )1

k k k

c k kk

c k

Log frequency

log ( )k

At low frequencies 2( )k k

2k

At high frequencies 1 2( )k k k

1 2k k

1k

c


Loss factor

2

2 1 2

2

1 1

( ) c

k k k

c k k

Log frequency

log ( )

1k

c

At low frequencies 2

( )c

k

At high frequencies 1 2

1 1

1( )

c k k

k k


10 -2

10 -1

10 0

10 1

10 2

10 3

10 4

Transition region

Glassy

region

Loss factor

Stiffness

Stiffn

ess a

nd

lo

ss fa

cto

r

Increasing frequency

Increasing temperature

Rubbery

region


Log frequency Log frequency

E o

r G

o

r

(a) (b)

Log frequency Log frequency

Visco-elastic materials – reduced

frequency nomogram

frequency

f, (

Hz)

modulu

s E

and loss f

acto

r η

reduced frequency fαT, Hz

temperature


frequency nomogram

• If the effects of damping and temperature on the damping behaviour

of materials are to be taken into account, then the temperature-

frequency equivalence (reduced frequency) is important.

• E and η are plotted against the reduced frequency parameter, fαT.

• To use the nomogram, for each specific f and Ti move down the Ti

line until it crosses the horizontal f line. The intersection point X,

corresponds to the value of fαT. Then move vertically up to read

the values of E and η.


frequency nomogram

Visco-elastic materials and damping

treatments

Modulu

s o

r lo

ss f

acto

r

Temperature

• For damping treatments visco-elastic materials should generally be used in their

transition region, where the loss factor is highest. However small changes in

temperature can have a large change in stiffness.

Useful range for

damping treatments

Typical material properties

Material

Young’s

modulus

E (N m-2)

Density

ρ (kg m-3)

Poisson’s

ratio ν

Loss factor

η

Mild Steel

2e11

7.8e3

0.28

0.0001 – 0.0006

Aluminium Alloy

7.1e10

2.7e3

0.33

0.0001 – 0.0006

Brass

10e10

8.5e3

0.36

< 0.001

Copper

12.5e10

8.9e3

0.35

0.02

Glass

6e10

2.4e3

0.24

0.0006 – 0.002

Cork

1.2 – 2.4e3

0.13 – 0.17

Rubber

1e6 - 1e9

≈ 1e3

0.4 – 0.5

≈ 0.1

Plywood*

5.4e9

6e2

0.01

Perspex

5.6e9

1.2e3

0.4

0.03

Light Concrete*

3.8e9

1.3e3

0.015

Brick*

1.6e10

2e3

0.015

Damping treatments using visco-

elastic materials

Viscoelastic damper

Viscoelastic

material

Damping treatments using visco-elastic

materials

Unconstrained layer damping

h2

h1

(1 )dE j

sE

• Applied to one free surface

• Loss due to extension of damping material (E product is

important)

• Frequency independent (apart from material properties)

Undeformed structure Deformed structure

Structure

Damping material


h2

h1

(1 )dE j

sE

Undeformed structure Deformed structure

Structure

Damping material

• The strain energy in a uniform beam of length l is given by

22

2

0

1

2

ld y

U EI dxdx

x

y

(1)


• A definition of the loss factor is

1 energy dissipated per cycle

2 maximum energy stored per cycle

Let loss factor for the structureb composite

Let loss factor for the d damping layer

Assume that the loss factor in the structure is negligible compared

with the damping layer

energy dissipated per cycle in the structure

maximum energy stored per cycle in the structureb

d

composite

composite

energy dissipated per cycle in the

maximum energy stored per cycle in the

damping layer

damping layer


So maximum energy stored per cycle in the

maximum energy stored per cycle in the struct u

reb

d composit

dampin l

e

g ayer

Substitute from (1) gives

b d d

d d d s s

E I

E I E I

(I’s are about the composite neutral axis)

Thin damping layers

b d d

d s s

E I

E I

Thus it is not sufficient to have a large ηd; a large Edηd is also required

Thick damping layers

b d

To be effective damping layers of similar thickness to the structure are

required Weight penalty


10 -1

10 0

10 1

10 2

10 -3

10 -2

10 -1

10 0

c

b

d

2

1

h

h

1

1

b

s sd

d d

E I

E I

Ed/Es=10-1

Ed/Es=10-2

Ed/Es=10-3

Ed/Es=10-4

i.e. require:

Ed large

h2 large

d large

In this region the

damping layer

also affects the

overall stiffness

of the structure.

Constrained layer damping

• Sandwich construction

• Loss due to shear of damping layer (G product is important)

• Constraining layers must be stiff in extension

• Thin damping layers can give high losses

h3

h2

h1

d

(1 )G j

Damping material Constraining layer

Undeformed

constrained layer

Deformed

constrained layer

Structure E1

E3

• Frequency dependent properties


2

1 1 3 32

1 1

b

Gg

E h E hh k

Bending wavenumber

2

1 1 3 3

1 1 3 3 1 1 3 3

( )E h E h dL

E h E h E I E I

composite/ depends on two non-dimensional parameters:

• ‘geometric parameter’ (also depends on elastic moduli)

• here I1 and I3 are relative to own centroids

• large for deep core, small for thin core (d)

• ‘shear parameter’

• large for low frequencies, small for high frequencies

h1 : h1 : h3 L

1 : 0.1 : 0.1 0.46

1 : 0.1 : 1 3.63

1 : 1 : 1 12

1 : 10 : 1 363

(E1=E3)


10 -2

10 -1

10 0

10 1

10 2

10 -4

10 -3

10 -2

10 -1

10 0

L=100

L=10

L=0.1

L=1

2

1 1 3 32

1 1

b

Gg

E h E hh k

2

1 1 3 3

1 1 3 3 1 1 3 3

( )E h E h dL

E h E h E I E I

Shear parameter

composite

In this region

large L means

much stiffer

structure


At low frequencies (high temp) constrained layer damping is

ineffective because the damping material is operating in its rubbery

region and is thus soft. The structure and the constraining layer

become uncoupled.

Effectiveness also depends on loss factor of damping material

At high frequencies (low temp) constrained layer damping is

ineffective because the damping material is operating in its glassy

region and is thus hard. The structure and the constraining layer tend

to move as one.

Other types of damping

Damping by air pumping

j tFe

Viscous damping

– reduces in effectiveness

at high frequencies

Pumping along joints

Squeeze-film damping I

Main plate

Attached plate

Thin layer of air

Damping is achieved by air being pumped between the main plate and

the attached plate (JSV 118(1), 123-139, 1987)

Squeeze-film damping II

Stationary casing

Rotating shaft

Ball/roller bearing

Oil film

Oil supply

vibration

• Oil film is squeezed between the bearing and the housing

generating damping forces

Summary: means of adding damping

• unconstrained layer damper

• constrained layer damper (can be multiple layers)

• squeeze film damping

• friction damping: important at bolted or rivetted joints.

• impact damper

• change in structural material.

• tuned vibration absorber (added damped mass-spring systems)

Measurement of Damping

• Decay of free vibration measurement

• Response curve at resonance measurement

• Complex modulus measurement

Measurement of Damping –decay of

free vibration

ntx Xe

t

x t

2

2 2

1d

d n

T

1t 2 1 dt t T

1x

2x

1

2

n dTxe

x

Measure the amplitude

of two successive cycles

1

2

ln n d

xT

x

Logarithmic decrement

So 2

2

1

For light damping 2

So 2

measured

Measurement of Damping – decay of

free vibration

• In practice it is better to measure the logarithmic decrement

over a number of cycles

1ln i

i n

x

n x

n can be determined by knowledge of the natural frequency

and the time between xi and xi+n

Measurement of Damping – response

curve at resonance

1 2n

X

F

maxX

F

max

2

X

F

frequency

Response curve

ReX

F

ImX

F

n

2

1 k

12

Nyquist plot

2 1

2 n

Measurement of Damping – complex

modulus measurement

shaker

specimen

area A

very stiff, light material

Impedance head

(measures force and acceleration)

l

X

F

(1 )k jMeasure dynamic stiffness (1 )

Fk j

X

Stiffness ReF

kX

Loss factor

Im

Re

F

X

F

X

Compute E by using EA

kl

Summary

• Review of damping mechanisms

• Modelling and characteristics of viscoelastic

damping

• Application of damping treatments

• Measurement of damping

References

• A.D. Nashif, D.I.G. Jones and J.P. Henderson, 1985, Vibration

Damping, John Wiley and Sons Inc.

• C.M. Harris, 1987, Shock and Vibration Handbook, Third

Edition, McGraw Hill.

• L.L. Beranek and I.L. Ver, 1992. Noise and Vibration Control

Engineering, John Wiley and Sons.

• D.J. Mead, 1999, Passive Vibration Control, John Wiley and

Sons.

• F.J. Fahy and J.G. Walker, 2004, Advanced Applications in

Acoustics, Noise and Vibration, Spon Press (chapter 12

Vibration Control by M.J. Brennan and N.S. Ferguson).

• F.S. Tse, I.E. Morse and R.T. Hinkle, 1978, Mechanical

Vibrations, Theory and Applications, Second Edition, Allyn and

Bacon, Inc.

Documents

Professor Mike Brennan - UNESP · Forced vibration – effect of damping Log frequency Log 1 k k m x c FejtZ Frequency Regions 2 2( ) 2 1 F k m cZ X F X ... Assume that the loss factor