Download Mechanical Mobility and Impedance - FEIS

Mechanical Mobility and
Impedance
Mike Brennan
UNESP, Ilha Solteira
São Paulo
Brazil
Mechanical Mobility and Impedance
• The response of a structure to a harmonic force can be expressed
in terms of its mobility or impedance
At frequency ω the velocity can
be written in complex notation
v(t) = Vejωt
F V
General linear
system
The mobility is defined as
Mobility =
V ( jω )
F ( jω )
V is the complex amplitude.
Similarly for the force f(t) = Fejωt
The impedance is defined as
Impedance =
F ( jω )
V ( jω )
• 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)
Accelerance =
Acceleration
Force
Velocity
Mobility =
Force
Displacement
Receptance =
Force
Apparent Mass =
Impedance =
Force
Acceleration
Force
Velocity
Force
Dynamic Stiffness =
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 jωt and x = Xe jωt
Also, block one end so that x2 = 0
k
F
So
F = kX
KV
V
then F =
Because X =
jω
jω
So the impedance of a spring is
given by
Zk =
F
k
=
V
jω
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 jωt and v = Ve jωt
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
xɺɺ
f1 + f2 = mxɺɺ
f2 = mxɺɺ − f1
• rigid
• force does not pass
through it unattenuated
ɺɺ jωt
Assume f = Fe jωt and xɺɺ = Xe
Also, set one end to be free
so that f2 = 0
F
m
ɺɺ
X
So
ɺɺ
F = mX
ɺɺ = jωV then F = j ω mV
Because X
So the impedance of a mass is
given by
Zm =
F
= jωm
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
• Spring Zk =
k
− jk
=
jω
ω
• Damper Zc = c
• Mass Zm = jω m
Zm
Zk
Zc
Re
Log |impedance|
Im
Zm
Zc
Zk
Log frequency
Mobilities of Simple Elements - Summary
• Spring Yk =
jω
k
• Damper Yc =
1
−j
1
Y
=
=
• Mass m
jωm ωm
c
Yk
Ym
Yc
Re
Log |mobility|
Im
Yk
Yc
|Ym|
Log frequency
Mobility and Impedance of Simple
Elements
• Can define ‘the impedance’ of a
spring in terms of the relative
velocity of the two ends.
• Can only define ‘the mobility’ of
a spring if one end is blocked.
• Can define ‘the impedance’ of a
damper in terms of the relative
velocity of the two ends.
• Can only define ‘the mobility’ of
a damper if one end is blocked.
• Can only define ‘the impedance’
of a mass if one end is free.
• Can define ‘the mobility’ of a
mass in terms of the sum of the
forces at the two ends.
(velocity at both ends is same).
(force at both ends is same).
Examples of impedance / mobility
mass
F = mXɺɺ Zmass = j ωm
− jk
spring
F = kX Z spring =
damper
F = cXɺ Z damper = c
infinite beam
infinite plate
ω
−j
Ymass =
ωm
jω
Yspring =
k
1
Ydamper =
c
1/ 4
3/ 4
Z beam = 2(1 + j )
ω 1/ 2(EI )(
ρA )
Z plate = 8h
2
Eρ
12(1 − ν 2 )
= 2.3c L ρh 2
Area A, second moment of area I, thickness h, Young’s modulus E, density ρ, Poisson’s
ratio ν
c L = E / ρ(1 − ν 2 )
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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.
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