阻抗与导纳

mass in terms of the sum of the
(velocity at both ends is same).
forces at the two ends.
Examples of impedance / mobility
mass spring damper
F mX Z mass j m
f2 k x2 x1
f1 f2
• no mass
• force passes through it unattenuated
x
So F kX
Because
XV
j
then F KV
j
So the impedance of a spring is
given by
Zk
F V
k
j
Assume f Fe jt and x Xe jt
Impedance
F V
j 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
• Point Mobility
Y11
1
jm k
c
j
k
j 2m
jc
Log |impedance| Log |mobility|
Frequency Response Functions
F
V
Fm
Fk
Fc
mk c
• Point Mobility
0.01
Stiffness line
0.1
• 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.
General linear system
V is the complex amplitude. Similarly for the force f(t) = Fejt
The mobility is defined as
Mobility
V F
j j
The impedance is defined as
F1
General linear
F2
system
V1
V2
• The two parameters at each input point are force, F, and velocity, V.
Simple Idealised Elements
• Spring
f1
k
f2
F
k
x1
x2
f1 k x1 x2
2.3cL h 2
Area A, second moment of area I, thickness h, Young’s modulus E, density , Poisson’s
ratio
cL E / (1 )2
12
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).
Impedances of Simple Elements - Summary
• Spring
Zk
k jk
j
Im
• Damper Zc c
• Mass Zm jm
Log |impedance|
Zm
Zc Re
Zm Zc
Zk
Zk
Log frequency
Mobilities of Simple Elements - Summary
• The response of a structure to a harmonic force can be expressed in terms of its mobility or impedance
FV
At frequency the velocity can
be written in complex notation v(t) = Vejt
高等结构动力学 阻抗与导纳
Vibration control
Vibration Problem
Understand problem
Modelling (Mobility and Impedance Methods)
Solve Problem
Measurement
Mobility and Impedance
• Spring
Yk
j
k
Im
•
Damper Yc
1 c
• Mass Ym
1
jm
j
m
Yk
Yc Re
Yk Yc
Log |mobility|
Ym
|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.
• Adding Elements in Parallel
Mass-less
Rigid link
F
V kc
• Same velocity • Shared force
F1 Z1V F2 Z2V F F1 F2
F Z1 Z2 V
N
Ztotal Z j j 1
Connecting Simple Elements
Mass-less
F Rigid link
mk c
Connecting Simple Elements
• Adding Elements in Parallel
F
F Fm Fk Fc
V
Fm
Fk
Fc
mk c
• Point Impedance
Z11
jm
k
j
c
At low frequency stiffness dominates At resonance damping dominates At high frequency mass dominates
• 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
Simple Idealised Elements
• Mass
f1
f2
m
x
F
m
X
So F mX
f1 f2 mx
Because X jV then F jmV
f2 mx f1
• rigid • force does not pass
through it unattenuated
So the impedance of a mass is given by
• Adding Elements in Series
F
k
c
V
V1
V2
• Same force • Shared velocity
V1 Y1F V2 Y2F V V1 V2
V Y1 Y2 F
N
Ytotal Yj j 1
Connecting Simple Elements
• Adding Elements in Parallel
• Impedance/mobility of a finite structure tends to that of the equivalent infinite structure at high frequency and/or high damping.
13
Connecting Simple Elements
Assume f Fe jt and v Ve jt Also, block one end so that v2 0
So the impedance of a damper is given by
Zc
F V
c
Note that the force is in phase with the velocity. Thus a damper is a resistive element that dissipates energy
Simple Idealised Elements
• Viscous damper
f1
c
f2
F
c
V
v1
v2
So F cV
f1 c v1 v2 f2 c v2 v1
f1 f2
• no mass or elasticity • force passes through
it unattenuated
• Can only define ‘the mobility’ of a damper if one end is blocked.
• Can only define ‘the impedance’ • Can define ‘the mobility’ of a
of a mass if one end is free.
F kX
Z spring
jk
F cX Z damper c
Ymass
j m
Yspring
j k
Ydamper
Baidu Nhomakorabea
1
c
infinite beam
Z beam 2(1 )j ( 1/2)( EI )1/ 4 A 3/ 4
infinite plate
Z plate 8h 2
E 12(1 2)
Note that the force is in quadrature with the velocity. Thus a spring is a reactive
Also, block one end so that x2 0 element that does not dissipate energy
Zm
F V
jm
Assume f Fe jt and x Xe jt
Also, set one end to be free so that f2 0
Note that the force is in quadrature with the velocity. Thus a mass is a reactive element that does not dissipate energy
• Impedances
N
Ztotal Z j j 1
• Mobilities
1
N
1
Y Y total j 1 j
• Example - SDOF System
F m kc
V
V
Note that this representation indicates that one end of the mass is connected to an inertial reference point (F=0)
Note that both mobility and impedance are frequency domain quantities
Frequency Response Functions (FRFs)
Accelerance = Acceleration Force
Apparent Mass = Force Acceleration
Mobility = Velocity Force
Impedance = Force Velocity
Receptance = Displacement Force
Dynamic Stiffness = Force Displacement
Mobility and Impedance Methods
• Can only define ‘the mobility’ of a spring if one end is blocked.
(force at both ends is same).
• Can define ‘the impedance’ of a damper in terms of the relative velocity of the two ends.