modal testing and analysis
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1
Modal Testing and Analysis
Saeed Ziaei-Rad
1389بھمن 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
mkXmkkXXm
Xetxkxxm
ti
Spatial Model (Free vibration)
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Modal Analysis and Testing 4
Undamped Systems (Forced vibration)
2
2
1)()(
)()()(
)(
mkFXH
FXmkFetfXetx
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(
)
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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
kcsmsXetx
kxxcxm
Oscillatory solution
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Modal Analysis and Testing 6
Viscous Damping (Forced Vibration)
mkcgArcH
cmkH
cimkFXH
FeXekcimtfkxxcxm
titi
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)
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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
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Modal Analysis and Testing 8
Structural Damping
ikH
idmkH
FXkidm
FXkicm
tfkxxcxmdc
e
e
e
))/(1(/1)(
)(1)(
)(
)(
)(/
20
2
2
2
Equivalent Viscous damping
=Structural damping loss factor
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Modal Analysis and Testing 9
Alternative Forms of FRF
)(FX
FeXeH ti
ti
Receptance
)(FV
FeVeY ti
ti
Mobility
)(FA
FeAeA ti
ti
Inertance or accelerance
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Modal Analysis and Testing 10
Relation between receptance and mobility
90) ()(
) ( )(
)()()(
HY
titi
ti
HY
HiFXi
FVY
VeXeitxtvXetx
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Modal Analysis and Testing 11
Relation between receptance and Inertance
180) ()(
) ( )(
)()()(
2
22
2
HA
titi
ti
HA
HF
XFAA
AeXetxtaXetx
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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
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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)
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Modal Analysis and Testing 14
Modulus vs. Frequency
Receptance FRF Mobility FRF
Inertance FRF1389بھمن
Modal Analysis and Testing 15
Modulus vs. FrequencyK=100000
K=1000000
M=10
M=100
Receptance FRF Mobility FRF
Inertance FRF1389بھمن
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
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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
mm
loglog/
mmi
mmlog/1
kklog
/1
kki
loglog/
kk
loglog2/2
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Modal Analysis and Testing 18
Real and Imaginary vs. Frequency
Receptancd FRF Mobility FRF
Inertance FRF
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Modal Analysis and Testing 19
Real vs. Imaginary (Viscous Damping)
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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(
)()()()()(
cVcUYVYU
Letcmk
mkY
cmkcY
cmkmkic
cimkiHiY
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Modal Analysis and Testing 21
Real vs. Imaginary (Structural Damping)
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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)(
ddVUYVYU
Letdmk
dY
dmkmkY
dmkidmk
idmkH
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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.
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