1 Modal Testing and Analysis Saeed Ziaei-Rad. Modal Analysis and Testing2 Single Degree-of-Freedom...
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Transcript of 1 Modal Testing and Analysis Saeed Ziaei-Rad. Modal Analysis and Testing2 Single Degree-of-Freedom...
1
Modal Testing and Analysis
Saeed Ziaei-Rad
Modal Analysis and Testing 2
Single Degree-of-Freedom (SDOF)
UndampedViscously DampedHysterically (Structurally) Damped
Modal Analysis and Testing 3
Undamped Systems (Theory)
k
m
F(t)
2/10
2
2
)/(
0)(
0
)(
0
mk
Xmk
kXXm
Xetx
kxxmti
Spatial Model (Free vibration)
Modal Analysis and Testing 4
Undamped Systems (Forced vibration)
2
2
1)()(
)(
)(
)(
)(
mkF
XH
FXmk
Fetf
Xetx
tfkxxm
ti
ti
FRF=Frequency
Response Function
0 2 4 6 8 10 12 14 16 18 200
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
(Rad/s)
H(
)
Modal Analysis and Testing 5
Viscous Damping (Free Vibration)
200
20
2002,1
2
1,
)(
)2/(, /
1
0
)(
0
d
tit
st
deXetx
kmcmk
is
kcsms
Xetx
kxxcxm
Oscillatory solution
6Modal Analysis and Testing
Viscous Damping (Forced Vibration)
mk
cgArcH
cmkH
cimkF
XH
FeXekcim
tfkxxcxmtiti
2
222
2
2
tan)(
)()(
1)(
)(
1)(
)(
)(
0 2 4 6 8 10 12 14 16 18 200
0.01
0.02
0.03
0.04
0.05
0.06
0.07
0.08
0.09
0.1
(Rad/s)
H(
)
0 2 4 6 8 10 12 14 16 18 200
20
40
60
80
100
120
140
160
180
(Rad/s)
(D
egre
e)
Modal Analysis and Testing 7
Structural Damping
x
f
x
f
x
f
# Viscous damping is not a good representative of real structures.
# Damping in real structures is frequency-dependent.
# A damper whose rate varies with frequency.
Viscous damper Dry friction Structural damping
Modal Analysis and Testing 8
Structural Damping
i
kH
idmkH
FXkidm
FXkicm
tfkxxcxm
dc
e
e
e
))/(1(
/1)(
)(
1)(
)(
)(
)(
/
20
2
2
2
Equivalent Viscous damping
=Structural damping loss factor
Modal Analysis and Testing 9
Alternative Forms of FRF
)(F
X
Fe
XeH
ti
ti
Receptance
)(F
V
Fe
VeY
ti
ti
Mobility
)(F
A
Fe
AeA
ti
ti
Inertance or accelerance
Modal Analysis and Testing 10
Relation between receptance and mobility
90
) ()(
) ( )(
)()(
)(
HY
titi
ti
HY
HiF
Xi
F
VY
VeXeitxtv
Xetx
Modal Analysis and Testing 11
Relation between receptance and Inertance
180
) ()(
) ( )(
)()(
)(
2
22
2
HA
titi
ti
HA
HF
X
F
AA
AeXetxta
Xetx
Modal Analysis and Testing 12
Definition of FRFs
Response Parameter: R
Standard FRF: R/F Inverse FRF: F/R
Displacement
ReceptanceAdmitanceDynamic FlexibilityDynamic Compliance
Dynamic Stiffness
Velocity Mobility Mechanical Impedance
Acceleration InertanceAccelerance
Apparent Mass
Modal Analysis and Testing 13
Graphical Display of FRFs
Modulus of FRF vs. frequency and phase vs. frequency (Bode type of plot)Real part of FRF vs. frequency and imaginary part vs. frequencyReal part of inverse FRF vs. frequency (or frequency^2) and imaginary part of inverse FRF vs. frequency (or frequency^2) Real part of FRF vs. imaginary part of FRF (Nyquist type of plot)
Modal Analysis and Testing 14
Modulus vs. Frequency
Receptance FRF
Mobility FRF
Inertance FRF
Modal Analysis and Testing 15
Modulus vs. Frequency
K=100000
K=1000000
M=
10
M=
100
K=100000
K=100000
0
M=10
M=100
M=10
M=100
K=100000
K=1000000
Receptance FRF Mobility FRF
Inertance FRF
Modal Analysis and Testing 16
Modulus vs. Frequency
A low Frequency straight-line (correspond to stiffness)A high frequency straight-line (correspond to mass)The resonant region with its abrupt magnitude and phase variation
Modal Analysis and Testing 17
Frequency Response of Mass and stiffness Elements
FRF Mass Stiffness
)(log
)(
H
H
)(log
)(
Y
Y
)(log
)(
A
A
log2log
/1 2
m
m
loglog
/
m
mi
m
m
log
/1
k
k
log
/1
k
ki
loglog
/
k
k
loglog2
/2
Modal Analysis and Testing 18
Real and Imaginary vs. Frequency
Receptancd FRF
Mobility FRF
Inertance FRF
Modal Analysis and Testing 19
Real vs. Imaginary (Viscous Damping)
Modal Analysis and Testing 20
Real vs. Imaginary (Viscous Damping)
222
222
2
222
2
222
22
2
)2/1()2/1(
)Im(
)Re(
)()(
)()Im(
)()()Re(
)()(
)()()(
cVcU
YV
YU
Let
cmk
mkY
cmk
cY
cmk
mkic
cimk
iHiY
Modal Analysis and Testing 21
Real vs. Imaginary (Structural Damping)
Modal Analysis and Testing 22
Real vs. Imaginary (Structural Damping)
222
222
222
2
222
2
2
)2/1()2/1(
)Im(
)Re(
)()()Im(
)()(
)()Re(
)()(
)(1)(
ddVU
YV
YU
Let
dmk
dY
dmk
mkY
dmk
idmk
idmkH
Modal Analysis and Testing 23
Conclusions
Close inspection of real structures suggests that viscous damping is not a good representative for MDOF systems.All structures show a degree of structural damping.Structural damping acts like an imaginary stiffness in frequency domain. Modulus vs. Frequency and Nyquist type plots for FRFs are more common.
