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
dissipation of mechanical energy (usually conversion into heat)
• 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
 Review of Damping Mechanisms
Viscous damping

m
x
k c

The equation of motion is

mx  cx  kx  0
inertia damping stiffness
force force force
Example - automotive suspension system
 Free vibration effect of damping
The underdamped displacement of the mass is given by

x  Xe nt
sin d t   
Exponential decay term Oscillatory term

 = Damping ratio = c  2mn  0    1

n = Undamped natural frequency = k m
d = Damped natural frequency = n 1   2
 = Phase angle
 Free vibration effect of damping

nt
x  Xe

time

d x  Xe nt
sin d t   
2   Damping ratio
Td 
d Td  Damping period
  Phase angle
 Hysteretic Damping
Fe jt Fe jt

m x m x
k 1  j  k c
Loss factor
Equation of motion
mx  k 1  j  x  Fe jt mx  cx  kx  Fe jt
Assuming a harmonic response x  Xe jt leads to
X 1 X 1
 
F k   2m  jk F k   2 m  j c
Setting responses to be equal at resonance gives
  2 where   c 2 mk

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

Viscous damping force is in-phase with and proportional to velocity
 Hysteretic Damping
Hysteretic damping model with a constant stiffness and loss factor
is only applicable for harmonic excitation
f f

m x m x
k c k 1  j 
causal acausal

f (t ) f (t )

t t
x (t ) x (t )

t t
 Coulomb Damping
Coulomb damping is caused by sliding or dry friction
x
x +ve Fd 
k f x -ve Fd 
m
Fd = friction force (always
opposes motion)
Fd
Fd   Fn x 0
0 x 0 Fn  mg is the reaction force
x
Fn x 0

• 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
 How to compare damping mechanisms?
Energy dissipated by a viscous damper
Now if
c x  X sin t   
Fd
then x   X cos t   

x Substitute into (1) gives

Fd  cx 2 

Energy lost per cycle is E  c 2 X 2  cos2 t    dt

0
E  work done = force  distance
which evaluates to give
T
dx
E   Fd dx   Fd dt
dt
0
E   c X 2
2 

E  
0
cx 2dt (1)
 Returning to Coulomb Damping
x
k f
m

• Since the damping force Fd = Fn is constant and the distance travelled
in one cycle is 4X, the energy dissipation per cycle is:

E  4Fn X

Noting that E   ceq X 2

4 Fn
Gives an equivalent viscous damping coefficient ceq 
 X

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

 Returning to Coulomb Damping

V
log
F
V 4F
F

log frequency

• Illustration of the non-linear effects of Coulomb damping

 Coulomb Damping

time

• Illustration of the non-linear effects of Coulomb damping

Equivalent viscous damping forces
and coefficients
Damping Damping Force Equivalent
Mechanism Viscous
Damping

Viscous cx c
k
Hysteretic k x

4  Fn
Coulomb  Fn
 X
 Forced vibration – effect of damping
Fe jt X 1

F (k  m 2 )2  (c )2
m x
Frequency Regions
k c
Low frequency  0  X / F  1/ k Stiffness controlled

Resonance 2  k m  X / F  1/ c Damping controlled

High frequency   n  X / F  1/ m Mass controlled

2 2

X
Log
F
1
k

Stiffness Damping Mass

controlled controlled controlled Log frequency
 Example: point mobility of 1.5 mm steel plate 0.6  0.5 m
0
10

 = 0.01
-1  = 0.1
Mobility (m/Ns) 10 infinite plate

-2
10

-3
10

-4
10

-5
10 1 2 3 4
10 10 10 10

Frequency (Hz)
• 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
k
Simple model
Voigt model
• Combination of
c
c k
Maxwell model

k2 Dynamic stiffness is
j t
Fe
k2  2  k1  k2  
 
k1  k1 
2
j
F c c
c k1  2
  
2
X  
1
Xe jt c k 
   1
 Returning to Complex Stiffness

k ( ) 1  j ( )  F
F  k ( ) 1  j ( ) 
X

F 
Stiffness determined from k ( )  Re  
X 

F 
Im  
Loss factor determined from  ( )  X 
F 
Re  
X 
Visco-elastic damping – material properties
Stiffness

k2  2  k1  k 2  At low frequencies k ( )  k2

c 2
k1  k1 
k ( ) 
 1
2
  
2
At high frequencies k ( )  k1  k2

c  k 
   1

k1  k2
log k ( )

k2

k1 Log frequency
c
Visco-elastic damping – material properties
Loss factor
 c
At low frequencies  ( ) 
c k2
 ( ) 
k2  2  k1  k 2 
 1
c 2
k1  k1  At high frequencies  ( ) 
c  k1  k2 
k1  k1 

log ( )

k1 Log frequency
c
 Visco-elastic materials
4
10
Glassy
Rubbery region
3
10 Transition region
region
Stiffness and loss factor

2
10
Stiffness

1
10

0
10

Loss factor
-1
10

-2
10

Increasing frequency
Increasing temperature
Visco-elastic materials

or 
E or G

(a) (b)

Log frequency
Log frequency Log
Logfrequency
frequency
Visco-elastic materials – reduced
frequency nomogram
temperature
modulus E and loss factor η

frequency f, (Hz)
reduced frequency fαT, Hz
 Visco-elastic materials – reduced
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.
Visco-elastic materials – reduced
frequency nomogram
 Visco-elastic materials and damping
treatments

Modulus or loss factor

Useful range for
damping treatments

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.
 Typical material properties

Material Young’s Density Poisson’s Loss factor

modulus ρ (kg m-3) ratio ν η
E (N m-2)
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
Damping treatments using visco-elastic
materials

Viscoelastic damper

Viscoelastic
material
 Unconstrained layer damping

Damping material Ed (1  j )
h2
h1 Structure E
s

Undeformed structure Deformed structure

• Applied to one free surface

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

important)

• Frequency independent (apart from material properties)

 Unconstrained layer damping
y

x
Damping material Ed (1  j )
h2

h1 Structure E
s

Undeformed structure Deformed structure

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

2
1
l
d y 
2
U EI   2  dx (1)
2 0  dx 
 Unconstrained layer damping
• A definition of the loss factor is

1 energy dissipated per cycle

 
2 maximum energy stored per cycle

Let b  loss factor for the composite structure

Let d  loss factor for the damping layer

Assume that the loss factor in the structure is negligible compared

with the damping layer

b energy dissipated per cycle in the composite structure


d maximum energy stored per cycle in the composite structure

energy dissipated per cycle in the damping layer


maximum energy stored per cycle in the damping layer
 Unconstrained layer damping
b maximum energy stored per cycle in the damping layer
So 
d maximum energy stored per cycle in the composite structure
Substitute from (1) gives
b Ed I d
 (I’s are about the composite neutral axis)
 d Ed I d  Es I s

Thin damping layers

 b Ed Id

 d Es Is
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
 Unconstrained layer damping
0
10

In this region the

damping layer
also affects the
Ed/Es=10-1 overall stiffness
-1
10 of the structure.
b
c

d
Ed/Es=10-2 b 1
 i.e. require:
-2
10  d 1  Es I s Ed large
Ed/Es=10-3 Ed Id h2 large
d large
Ed/Es=10-4
-3
10 -1 0 1 2
h2
10 10 10 10
h1
 Constrained layer damping
Constraining layer Damping material
G(1  j  )
E3
h3
d h2
h1 Structure E1
Undeformed Deformed
constrained layer constrained layer

• 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

• Frequency dependent properties

 Constrained layer damping
composite/ depends on two non-dimensional parameters:

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

L
 E1h1  (E3 h3 )d 2 h1 : h1 : h3
1 : 0.1 : 0.1
L
0.46
 E1h1  E3h3  E1I1  E3I3  1 : 0.1 : 1 3.63
1: 1 :1 12
• here I1 and I3 are relative to own centroids 1 : 10 : 1 363
• large for deep core, small for thin core (d) (E1=E3)

• ‘shear parameter’
G  1 1 
g 2 
 
h2kb  1 1 E3h3 
E h

Bending wavenumber

• large for low frequencies, small for high frequencies

 Constrained layer damping
0
10
In this region
L=100 large L means
-1
much stiffer
10 L=10 structure
composite L=1
 10
-2

L=0.1

10
-3

L
 E1h1  (E3 h3 )d 2
 E1h1  E3h3  E1I1  E3I3 
-4
10 -2 -1 0 1 2
10 10 10 10 10

G  1 1 
Shear parameter g  2 
 
h2kb  1 1 E3h3 
E h
 Constrained layer damping

Effectiveness also depends on loss factor of damping material

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.

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

Viscous damping
– reduces in effectiveness
at high frequencies

Fe jt
Pumping along joints
 Squeeze-film damping I
Attached plate
Thin layer of air Main plate

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

Oil supply

Stationary casing
Oil film

vibration Rotating shaft

Ball/roller bearing

• 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)

• friction damping: important at bolted or rivetted joints.

• squeeze film damping

• 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
x t 
nt Measure the amplitude
x  Xe of two successive cycles
x1
 enTd
x1 x2
x2
t  x1 
  ln    nTd
 x2 
Logarithmic decrement
2
So 
t1 t2  t1  Td 1  2
For light damping   2
2 2 
Td   measured
d n 1   2 So  
2
 Measurement of Damping – decay of
free vibration

• In practice it is better to measure the logarithmic decrement

over a number of cycles

1  xi 
  ln  
n  xi  n 

n can be determined by knowledge of the natural frequency

and the time between xi and xi+n
 Measurement of Damping – response
X
curve at resonance
F Response curve Nyquist plot
X max 1k

F  2 X 
Re  
F 
X max
2F 2 1

n X 
Im  
F 
1
n  2 frequency

2  1
 
2n
 Measurement of Damping – complex
modulus measurement
shaker
Impedance head
(measures force and acceleration)
very stiff, light material
specimen
l
area A

F
F
X Measure dynamic stiffness X  k (1  j )
k (1  j )
F 
Im  
F    X 
Stiffness k  Re   Loss factor
X  F 
Re  
EA X 
Compute E by using k 
l
 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.