Mobility and Impedance Methods
Professor Mike Brennan
Vibration control
Vibration Problem
Understand problem
Solve Problem
Modelling
(Mobility and
Impedance Methods)
Measurement
Mobility and Impedance
• The response of a structure to a harmonic force can be expressed
in terms of its mobility or impedance
At frequency  the velocity can
be written in complex notation
v(t) = Vejt
F V
General linear
system
The mobility is defined as
Mobility 
V  j 
F  j 
V is the complex amplitude.
Similarly for the force f(t) = Fejt
The impedance is defined as
Impedance 
F  j 
V  j 
• If the force and velocity are at the same point this is a ‘point’ mobility
• If they are at different points it is a ‘transfer’ mobility
Note that both mobility and impedance are frequency domain quantities
Frequency Response Functions (FRFs)
Acceleration
Accelerance =
Force
Force
Apparent Mass =
Acceleration
Velocity
Mobility =
Force
Force
Impedance =
Velocity
Receptance =
Displacement
Force
Dynamic Stiffness =
Force
Displacement
Mobility and Impedance Methods
• The total response of a set of coupled components can be expressed
in terms of the mobility of the individual components
• In the simplest case each component has two inputs (one at each end)
which permit coupling
F1
F2
General linear
system
V1
V2
• The two parameters at each input point are force, F, and velocity, V.
Simple Idealised Elements
• Spring
k
f1
x1
f2
x
x2
f1  k  x1  x2 
f2  k  x2  x1 
f1  f2
• no mass
• force passes through
it unattenuated
Assume f  Fe jt and x  Xe jt
Also, block one end so that x2  0
k
F
So
F  kX
KV
V
then F 
j
j
So the impedance of a spring is
given by
Because X 
Zk 
F
k

V
j
Note that the force is in quadrature with the
velocity. Thus a spring is a reactive
element that does not dissipate energy
Simple Idealised Elements
• Viscous damper
c
f1
v1
c
F
f2
v2
f1  c v1  v 2 
f2  c v 2  v1 
f1  f2
• no mass or elasticity
• force passes through
it unattenuated
Assume f  Fe jt and v  Ve jt
Also, block one end so that v 2  0
V
So
F  cV
So the impedance of a damper is
given by
Zc 
F
c
V
Note that the force is in phase with the
velocity. Thus a damper is a resistive
element that dissipates energy
Simple Idealised Elements
• Mass
f1
f2
m
x
f1  f2  mx
f2  mx  f1
• rigid
• force does not pass
through it unattenuated
Assume f  Fe jt and x  Xe jt
Also, set one end to be free
so that f2  0
F
m
X
So
F  mX
Because X  jV then F  jmV
So the impedance of a mass is
given by
F
Zm   j  m
V
Note that the force is in quadrature with the
velocity. Thus a mass is a reactive
element that does not dissipate energy
Impedances of Simple Elements - Summary
k
 jk

• Spring Zk 
j

• Damper Zc  c
• Mass Zm  jm
Zm
Zk
Zc
Re
Log |impedance|
Im
Zm
Zc
Zk
Log frequency
Mobilities of Simple Elements - Summary
1
j
1
Y


• Damper Yc 
• Mass m
jm m
c
j
• Spring Yk 
k
Yk
Ym
Yc
Re
Log |mobility|
Im
Yk
Yc
|Ym|
Log frequency
Examples of impedance / mobility
mass
F  mX Z mass  j m
 jk


spring
F  kX Z spring
damper
F  cX Z damper  c
j
Ymass 
m
j
Y spring 
k
1
Y damper 
c
infinite beam
Z beam  21
(  )j (1/)(2 EI) 1/ 4  A
infinite plate
Z plate  8h
2
E
121
(   2)
3/ 4
 2.3c L h 2
Area A, second moment of area I, thickness h, Young’s modulus E, density , Poisson’s
ratio 
c L  E / 1
(  )2
11
Notes on impedance / mobility
•
Real part of point impedance (or mobility) is always
positive (dissipation). Imaginary part can be positive
(mass-like) or negative (spring-like).
•
Infinite plate impedance is real and independent of
frequency (equivalent to a damper).
•
Beam impedance has a damper part and a mass part,
both frequency dependent.
•
Impedance/mobility of a finite structure tends to that
of the equivalent infinite structure at high frequency
and/or high damping.
12
Connecting Simple Elements
• Adding Elements in Parallel
Mass-less
Rigid link
k
F
V
c
• Same velocity
F1  Z1V
• Shared force
F  F1  F2
F  Z1  Z2 V
N
Ztotal   Z j
j 1
F2  Z2V
Connecting Simple Elements
• Adding Elements in Series
F
k
c
V1
V2
V
• Same force
• Shared velocity
V1  Y1F
V2  Y2F
V  V1  V2
V  Y1  Y2 F
N
Ytotal  Y j
j 1
Connecting Simple Elements
• Adding Elements in Parallel
• Impedances
• Mobilities
N
1
1

Ytotal j 1 Y j
N
Ztotal   Z j
j 1
• Example - SDOF System
F
m
k
V

F
Mass-less
Rigid link
V
c
Note that this representation
indicates that one end of the
mass is connected to an
inertial reference point (F=0)
m
k
c
Connecting Simple Elements
• Adding Elements in Parallel
F  Fm  Fk  Fc
F
V
Fm
m
Fk
k
Fc
c
At low frequency stiffness dominates
At resonance damping dominates
At high frequency mass dominates
• Point Impedance
k
Z11  jm 
c
j
• Point Mobility
1
Y11 
k
jm 
c
j
j

k   2 m  j c
Frequency Response Functions
F
m
Fk
k
c
• Point Impedance
Log |impedance|
  0.01
Fc
Log |mobility|
Fm
V
• Point Mobility
Stiffness
line
  0.1
Mass line
Log frequency
Stiffness
line
Mass line
  0.1
  0.01
Log frequency
Connecting Simple Elements
• Adding Elements in Series
• Impedances
1
Ztotal
• Mobilities
N
N
1

j 1 Z j
• Example
Ytotal  Y j
j 1
k
F
V
• Point Mobility
j 1
1
Y11 
 
k c jm
c
m
• Point Impedance
1
Z11 
j 1
1
 
k c jm
Connecting Simple Elements
• Adding Elements in Series
k
c
F
• Point mobility
At low frequency mass dominates
Log |mobility|
V
m
Mass
j line1
1
Y11 
 
k c jm
At resonance damping dominates
At high frequency stiffness dominates
1
c
Log frequency
Stiffness
line
Adding a Combination of Parallel and Series Elements
• Example - SDOF System

m
k
c
m
k
V
V
F
1

 1   1 
Z Z Z 
c 
 m  k
Log |impedance|
F
Ztotal
c
stiffness line
Damping line
Log frequency
mass
line
E.L.Hixson, Chapter 10 in Shock and Vibration Handbook
Coupling Together Complex
Arbitrary Systems
F1
F2
General linear
system
V1
V2
• An arbitrary system having only two inputs can be represented using
the sign convention shown in the figure above (F1 and F2 both act on
the element).
• When the system is not a simple mass, spring or damper it is necessary
to assign both point and transfer mobilities to a system to define
completely the inter-relationship between both inputs.
Mobility Method
F1
F2
General linear
system
V1
V2
• The equations describing the system are given by
V1  Y11F1  Y12F2
V2  Y21F1  Y22F2
which can be written in matrix form as
V1  Y11 Y12   F1 
 
 

V2  Y21 Y22  F2 
Mobility matrix
or
v  Yf
Mobility Method
V1  Y11 Y12   F1 
 
 

V2  Y21 Y22  F2 
V1
Y11 
F1 F 0
2
Force applied
at point 1
Point 2
Is free
Y11 and Y22 are point mobilities, which
relate the velocity at the point of excitation
to the force applied. They are defined as
V2
Y22 
F2
Force applied
at point 2
F1 0
Point 1
Is free
Y12 and Y21 are transfer mobilities, which relate the velocity at the point
of some remote point to the force applied. They are defined as
V1
Y12 
F2
F1 0
V2
Y21 
F1
F2 0
Mobility Method
Notes
• For a linear system Y12  Y21 because of reciprocity
• If the system is symmetric then Y11  Y22
• These mobilities are found by allowing one input to be free
• The mobility formulation is most useful in experimental work as the
point and transfer mobilities are easily measured
• Re{Y11} and Re{Y22} must be positive
Impedance Method
F1
F2
General linear
system
V1
V2
• The equations describing the system are given by
F1  Z11V1  Z12V2
F2  Z21V1  Z22V2
which can be written in matrix form as
 F1   Z11 Z12  V1 
 
 

F2  Z21 Z22  V2 
Impedance matrix
or
f  Zv
Impedance Method
 F1   Z11 Z12  V1 
 
 

F2  Z21 Z22  V2 
F1
Z11 
V1 V 0
2
Force applied
at point 1
Point 2
Is fixed
Z11 and Z22 are point impedances, which
relate the velocity at the point of excitation
to the force applied. They are defined as
Z22
F2

V2 V 0
1
Force applied
at point 2
Point 1
Is fixed
Z12 and Z21 are transfer impedances, which relate the velocity at the point
of some remote point to the force applied. They are defined as
F1
Z12 
V2 V 0
1
F2
Z21 
V1 V 0
2
Impedance Method
Notes
• For a linear system Z12  Z21 because of reciprocity
• If the system is symmetric then Z11  Z22
• These impedances are found by fixing all points except one
• The impedance formulation is most useful for theoretical formulation
and when experimentally working on light structures when blocking is
possible
• Re{Z11} and Re{Z22} must be positive
Impedance and Mobility matrices for simple elements
Spring
k  1 1
Zk 
j  1 1 
Yk is not defined as the element is massless,
and the mobilities are infinite if one input
is free
Damper
 1 1
Zc  c 


1
1


Yc is not defined as the element is massless,
and the mobilities are infinite if one input
is free
Mass
j
Ym 
m
1 1
1 1


Z m is not defined as a blocked output would
prevent motion
Coupling together complex arbitrary systems
F1
F2
F3
II
I
V1
V2
System I
V1  Y11F1  Y12F2
V2  Y21F1  Y22F2
V3
System II
V3  Y33F3
Coupling together complex arbitrary systems
- series connection
Rigid link
F1
F2
F3
II
I
V1
V2
When rigidly connected
System I
V1  Y11F1  Y12F2
V2  Y21F1  Y22F2
V3
F3  F2 (equilibrium of forces)
V3  V2 (continuity of motion)
System II
V3  Y33F3
Coupling together complex arbitrary systems
- series connection
Rigid link
F1
F2
F3
II
I
V1
V2
V3
Combining system equations gives the point and transfer mobilities of
the coupled system
V1
(Y21 )2
 Y11 
Point
F1
Y22  Y33
Transfer
Y12Y33
V2

F1 Y22  Y33
The natural frequencies of the coupled system occur when ImY22  Y33   0
The imaginary components embody the reactive elements which can
equal zero
Coupling together complex arbitrary systems
- parallel coupled system
For the uncoupled systems
F1
I
F
V1
II
V2
F2  Z22V2
When the systems are joined
by a rigid link
F2
V
F1  Z11V1
F  F1  F2
V  V1  V2
F
so
 Z11  Z22
V
The point mobility of the coupled system is given by
V
1
1


F Z11  Z22 1 Y11  1 Y22
Vibration source characterisation
Thévenin equivalent system
•
A vibration source connected to a load can be represented by a
blocked force Fb in parallel with an internal impedance Zi
connected to a load impedance Zl.
Fb
Fl
Load
Zl
Source
1
Fl  Fb
Zi
1
Zl
Zi
Vl
Fb
Vl 
Zi  Zl
Blocked force is the force generated by the source when it is connected to rigid load
Vibration source characterisation
Norton equivalent system
•
A vibration source connected to a load can be represented by the
free velocity of the source Vf in series with an internal impedance
Zi connected to a load impedance Zl.
Fl
Source
Zi
Vf
1
Vl  Vf
Zl
1
Zi
Load
Zl
Vl
Zl
Fl  Vf
Zl
1
Zi
Free velocity is the velocity of the source when the load is disconnected
Vibration source characterisation
•
Relationship between blocked force, free velocity and
internal impedance of the source
Thévenin equivalent system
Norton equivalent system
Fb
Vl 
Zi  Zl
Vl 
Fb
Zi
Vf
Zi  Zl
Vf
Fb = Zi Vf
b
Zi
Zi
Fl 
•
Zl
Fb
Zi  Zl
Similar equations involving mobilities.
Fl 
Zi Zl
Vf
Zi  Zl
b
Reduction of a system to Thévenin and
Norton equivalent systems – example
F
c2
c1
k2
k1
m1
m3
Z1  Zm1  Zk 1  Zc1
Z2  Zk 2  Zc 2
Z3  Zm3
F
F
Z2
b
b
Z1
Z3
Z2
Fb b
Z1
Blocked Force
Z2
Fb 
F
Z1  Z2
Z3 is not included because its ends
are attached to two rigid walls
Reduction of a system to Thévenin and
Norton equivalent systems – example
F
c2
c1
k2
k1
F
F
Z2
b
Z1
b
m1
Z1
Z3
m3
Z2
Fb b
Free velocity
Z2
Z1
Fb
Vf 
Zs
Source impedance
at connection point
b
Z3
Internal impedance
1
Z s  Z3 
1/ Z1  1/ Z2
Reduction of a system to Thévenin and
Norton equivalent systems – example
F
c2
c1
Fb
Blocked Force
b
k2
k1
b
Z2
Fb 
F
Z1  Z2
Zs
m1
m3
Z1  Zm1  Zk 1  Zc1
Vf
Zs
b
Fb
Vf 
Zs
Z2  Zk 2  Zc 2
Z3  Zm3
Free velocity
Internal impedance
1
Z s  Z3 
1/ Z1  1/ Z2
Summary
• Introduction to mobility and impedance
approach
• Lumped parameter systems
• Arbitrary systems
• Coupling of systems
• Source characterisation
References
• E.L. Hixson, 1997. Shock and Vibration Handbook (Chapter 10),
edited by C.M. Harris, Third Edition, McGraw Hill. Mechanical
Impedance.
• R.E.D. Bishop and D.C. Johnson, 1960. The Mechanics of
Vibration, Cambridge University Press.
• L. Cremer, M. Heckl and E.E. Ungar, 1990, Structure-Borne
Sound, Springer-Verlag, Berlin-Heidelberg-New York.
• L.L. Beranek and I.L. Ver, 1992. Noise and Vibration Control
Engineering, John Wiley and Sons.
• F.J. Fahy and J.G. Walker, 2004, Advanced Applications in
Acoustics, Noise and Vibration, Spon Press (chapter 9 Mobility
and impedance methods in structural dynamics by P. Gardonio
and M.J. Brennan).