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. viscoelastic 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
k
x
c
The equation of motion is
mx  cx  kx  0
inertia
force
damping
force
stiffness
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
c  2mn 
0  

= Damping ratio =
n
= Undamped natural frequency =
d

= Damped natural frequency =
= Phase angle
k m
n 1   2
 1
Free vibration effect of damping
x  Xe
nt
time

d
x  Xe
Td 
2
d
nt
sin d t   
  Damping ratio
Td  Damping period
  Phase angle
Hysteretic Damping
Fe jt
m
Fe jt
m
x
k 1  j 
k
x
c
Loss factor
Equation of motion
mx  k 1  j  x  Fe jt
Assuming a harmonic response
X
1

F k   2m  jk
mx  cx  kx  Fe jt
x  Xe jt leads to
X
1

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
k
m
x
c
k 1  j 
causal
f (t )
x
acausal
f (t )
t
t
x (t )
x (t )
t
t
Coulomb Damping
Coulomb damping is caused by sliding or dry friction
x
k
x +ve Fd 
x -ve Fd 
f
m
Fd = friction force (always
opposes motion)
Fd
Fd   Fn
x
x 0
0
x 0
Fn
x 0
Fn  mg is the reaction force
• 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
x  X sin t   
c
Fd
then x   X cos t   
x
Substitute into (1) gives
Fd  cx
E  c 2 X 2
Energy lost per cycle is
E  work done = force  distance
T
E 
 Fd dx   Fd
0
dx
dt
dt

0
cx 2dt

cos2 t    dt
0
which evaluates to give
E   c X 2
2 
E 
2 
(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
Gives an equivalent viscous damping coefficient
ceq 
4 Fn
 X
Note that this is a non-linear form of damping – dependent upon amplitude
Returning to Coulomb Damping
log
V
V
F
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
Mechanism
Viscous
Hysteretic
Coulomb
Damping Force
Equivalent
Viscous
Damping
cx
c
k x
k
 Fn

4  Fn
 X
Forced vibration – effect of damping
Fe jt
m
k
X
1

F
(k  m 2 )2  (c )2
x
Frequency Regions
c
Low frequency
Resonance
 0
2  k m
 X / F  1/ k
Stiffness controlled
 X / F  1/ c
Damping controlled
2
2
High frequency   n  X / F  1/ m Mass controlled
X
Log
F
1
k
Stiffness
controlled
Damping
Mass
controlled controlled
Log frequency
Example: point mobility of 1.5 mm steel plate 0.6  0.5 m
Mobility (m/Ns)
10
10
10
10
10
10
0
 = 0.01
 = 0.1
-1
infinite plate
-2
-3
-4
-5
10
1
10
2
10
3
10
4
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 nonlinear 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
Fe
c
k1
j t
Xe jt
Dynamic stiffness is
k2  2  k1  k2 



j
2
k1  k1 
c
F c

2
2
X


1

 
c k 
   1
Returning to Complex Stiffness
F
k ( ) 1  j ( ) 
F
 k ( ) 1  j ( ) 
X
x
F 
Stiffness determined from k ( )  Re  
X 
F 
Im  
X 
Loss factor determined from  ( ) 
F 
Re  
X 
Visco-elastic damping – material properties
Stiffness
k2  2  k1  k 2 

2
c
k1  k1 
k ( ) 
2
2



 1

c  k 
   1
At low frequencies k ( )  k2
At high frequencies k ( )  k1  k2
k1  k2
log k ( )
k2
k1
c
Log frequency
Visco-elastic damping – material properties
Loss factor

 ( ) 
c
k2  2  k1  k 2 

2
c
k1  k1 
At low frequencies  ( ) 
At high frequencies  ( ) 
c
k2
1
c  k1  k2 
k1  k1 
log ( )
k1
c
Log frequency
Visco-elastic materials
4
10
Stiffness and loss factor
3
10
Rubbery
region
Transition region
Glassy
region
2
10
Stiffness
1
10
0
10
Loss factor
-1
10
-2
10
Increasing frequency
Increasing temperature
E or G
or 
Visco-elastic materials
(a)
(b)
Log
frequency
Log frequency
Log
Logfrequency
frequency
Visco-elastic materials – reduced
frequency nomogram
frequency f, (Hz)
modulus E and loss factor η
temperature
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 temperaturefrequency 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
Modulus or loss factor
Visco-elastic materials and damping
treatments
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
Mild Steel
Young’s
modulus
E (N m-2)
Density
ρ (kg m-3)
Poisson’s
ratio ν
Loss factor
η
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
1.2 – 2.4e3
Cork
0.13 – 0.17
Rubber
1e6 - 1e9
≈ 1e3
0.4 – 0.5
≈ 0.1
Plywood*
5.4e9
6e2
Perspex
5.6e9
1.2e3
Light Concrete*
3.8e9
1.3e3
0.015
Brick*
1.6e10
2e3
0.015
0.01
0.4
0.03
Damping treatments using viscoelastic 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
Damping material
x
Ed (1  j )
h2
h1
Structure E
s
Deformed structure
Undeformed structure
•
The strain energy in a uniform beam of length l is given by
U
2
d y 
1
EI   2  dx
2 0  dx 
l
2
(1)
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
So
b
maximum energy stored per cycle in the damping layer

d maximum energy stored per cycle in the composite structure
Substitute from (1) gives
b
Ed I d

 d Ed I d  Es I s
(I’s are about the composite neutral axis)
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
overall stiffness
of the structure.
Ed/Es=10-1
c
b
d
-1
10
b
1

 d 1  Es I s
Ed/Es=10-2
-2
10
i.e. require:
Ed large
h2 large
Ed Id
Ed/Es=10-3
d large
Ed/Es=10-4
-3
10
-1
10
0
10
1
10
2
10
h2
h1
Constrained layer damping
Constraining layer
d
h3
h2
h1
E3
Structure
Damping material
G(1  j  )
E1
Undeformed
constrained layer
Deformed
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)
E1h1  (E3 h3 )d 2

L
 E1h1  E3h3  E1I1  E3I3 
• here I1 and I3 are relative to own centroids
• large for deep core, small for thin core (d)
•
h1 : h1 : h3
1 : 0.1 : 0.1
1 : 0.1 : 1
1: 1 :1
1 : 10 : 1
(E1=E3)
‘shear parameter’
G  1
1 
g


2 
E
h
h2kb  1 1 E3h3 
Bending wavenumber
• large for low frequencies, small for high frequencies
L
0.46
3.63
12
363
Constrained layer damping
0
10
In this region
large L means
much stiffer
structure
L=100
-1
10
composite

L=10
L=1
-2
10
L=0.1
-3
10
E1h1  (E3 h3 )d 2

L
 E1h1  E3h3  E1I1  E3I3 
-4
10 -2
10
-1
10
0
10
1
10
2
10
G  1
1 

Shear parameter g 

2 
E
h
h2kb  1 1 E3h3 
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
x t 
Measurement of Damping –decay of
free vibration
x  Xe
x1
Measure the amplitude
of two successive cycles
nt
x2
x1
 enTd
x2
 x1 
  ln    nTd
 x2 
t
Logarithmic decrement
So

t2  t1  Td
t1
For light damping
Td 
2
d

2
n 1  
2
So
2
1  2
  2

 
2
measured
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
X
F
Measurement of Damping – response
curve at resonance
Nyquist plot
Response curve
1k
X max
F
 2
X max
2
2F
1
n
1
n  2
frequency
2  1
 
2n
X 
Re  
F 
X 
Im  
F 
Measurement of Damping – complex
modulus measurement
shaker
Impedance head
(measures force and acceleration)
very stiff, light material
l
specimen
area A
F
k (1  j )
X
F
Measure dynamic stiffness X  k (1  j )
F 
Stiffness k  Re  
X 
EA
Compute E by using k 
l
F 
Im  
X 


Loss factor
F 
Re  
X 
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.