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Modal Analysis 1
ExperimentalModal Analysis
Copyright 2003Brel & Kjr Sound & Vibration Measurement A/SAll Rights Reserved
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Modal Analysis 2
SDOF and MDOF Models
Different Modal AnalysisTechniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis PostProcessing
f(t)
x(t)
kc
m
= ++ + +
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Modal Analysis 3
Simplest Form of Vibrating System
D
d = D sinnt
m
k
T
Time
Displacement
Frequency1
T
Period, Tn in [sec]
Frequency, fn= in [Hz = 1/sec]1Tn
Displacement
k
mn= 2 fn =
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Mass and Spring
time
m1
m
Increasing massreduces frequency
1
nmm
k2
+== nf
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Mass, Spring and Damper
Increasing dampingreduces the amplitude
time
m
k c1 + c2
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Modal Analysis 6
)()()()( tftKxtxCtxM =++ &&&
M = mass (force/acc.)
C = damping (force/vel.)
K = stiffness (force/disp.)
)t(x&&)t(x& )t(x
Acceleration Vector
Velocity Vector
Displacement Vector
)t(f Applied force Vector
Basic SDOF Model
f(t)
x(t)
kc
m
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Modal Analysis 7
SDOF ModelsTime and Frequency Domain
kcjmF
XH
++==
2
1
)(
)()(
H()F() X()
|H()|
1k
H()
0
90
180
0 = k/m
12m
1c
)()()()( tkxtxctxmtf ++= &&&
f(t)
x(t)
kc
m
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Modal Analysis 8
Modal Matrix
Modal Model
(Freq. Domain)
( )( )
( )
( )
( ) ( ) ( )
( )
( )
( ) ( )
( )
=
3
1
23
22
12111
3
2
1
F
HH
H
H
HHH
X
X
XX
nnn
n
n
X1
X2
X3X
4
F3H21
H22
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Modal Analysis 9
MDOF Model
0
-90
-180
2 1
1+2
1 2
1+2
Frequency
FrequencyPhase
Magnitude
m
d1+ d2
d1
dF
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Modal Analysis 10
Why Bother with Modal Models?
Physical Coordinates = CHAOS Modal Space = Simplicity
q1
1
1
20112
2
q21
20222
q31
20332
3
Bearing
Rotor
Bearing
Foundation
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Modal Analysis 11
Definition of Frequency Response Function
F X
f f f
H(f) is the system Frequency Response Function
F(f) is the Fourier Transform of the Input f(t)
X(f) is the Fourier Transform of the Output x(t)
F(f) H(f) X(f)
)f(F
)f(X)f(H =
H
H
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Modal Analysis 12
Benefits of Frequency Response Function
F(f) H(f) X(f)
Frequency Response Functions are properties of lineardynamic systems
They are independent of the Excitation Function
Excitation can be a Periodic, Random or Transient functionof time
The test result obtained with one type of excitation can beused for predicting the response of the system to any othertype of excitation
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Modal Analysis 13
Compliance Dynamic stiffness
(displacement / force) (force / displacement)
Mobility Impedance
(velocity / force) (force / velocity)
Inertance or Receptance Dynamic mass(acceleration / force) (force /acceleration)
Different Forms of an FRF
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Modal Analysis 14
Alternative Estimators
)()()(
fFfXfH =
)()(
)(1 fG
fGfH
FF
FX=
)()()(
2 fGfGfH
XF
XX=
213 )( HHG
G
G
G
fH FX
FX
FF
XX
==
2
1
2
2 *)(
H
H
G
G
G
G
GG
Gf
XX
FX
FF
FX
XXFF
FX ==
=
X(f)F(f) H(f)
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Modal Analysis 15
Which FRF Estimator Should You Use?
Definitions:
User can choose H1, H2 or H3 after measurement
Accuracy for systems with: H1 H2 H3
Input noise - Best -
Output noise Best - -
Input + output noise - - Best
Peaks (leakage) - Best -
Valleys (leakage) Best - -
Accuracy
)()(
)(1 fG
fGfH
FF
FX=)()(
)(2 fG
fGfH
XF
XX=FX
FX
FF
XX
G
G
G
GfH =)(3
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Modal Analysis 16
SDOF and MDOF Models
Different Modal AnalysisTechniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis PostProcessing
f(t)
x(t)
kc
m
= ++ + +
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Modal Analysis 17
Three Types of Modal Analysis
1. Hammer Testing
Impact Hammer taps...serial or parallel measurements
Excites wide frequency range quickly
Most commonly used technique
2. Shaker Testing
Modal Exciter shakes product...serial or parallel measurements
Many types of excitation techniques Often used in more complex structures
3. Operational Modal Analysis
Uses natural excitation of structure...serial or parallel measurements Cutting edge technique
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Modal Analysis 18
Different Types of Modal Analysis (Pros)
Hammer Testing Quick and easy
Typically Inexpensive
Can perform poor man modal as well as full modal
Shaker Testing
More repeatable than hammer testing
Many types of input available
Can be used for MIMO analysis Operational Modal Analysis
No need for special boundary conditions
Measure in-situ
Use natural excitation
Can perform other tests while taking OMA data
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Modal Analysis 19
Different Types of Modal Analysis (Cons)
Hammer Testing Crest factors due impulsive measurement
Input force can be different from measurement to measurement(different operators, difficult location, etc.)
Calibrated elbow required (double hits, etc.) Tip performance often an overlooked issue
Shaker Testing
More difficult test setup (stingers, exciter, etc.)
Usually more equipment and channels needed
Skilled operator(s) needed
Operational Modal Analysis
Unscaled modal model Excitation assumed to cover frequency range of interest
Long time histories sometimes required
Computationally intensive
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Modal Analysis 20
Frequency Response Function
Input
Output
Time(Response) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s] Time(Response) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s]
Time(Excitation) - Input
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N] Time(Excitation) - Input
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N]
FFT
FFTFrequency Domain
Autospectrum(Excitation) - Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1 ,6k
100u
1m
10m
100m
1
[Hz]
[N] Autospectrum(Excitation) - Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1 ,6k
100u
1m
10m
100m
1
[Hz]
[N]
Autospectrum(Response) - Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1 ,6k
1m
10m
100m
1
10
[Hz]
[m/s] Autospectrum(Response) - Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1 ,6k
1m
10m
100m
1
10
[Hz]
[m/s]
Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100m
10
[Hz]
[(m/s)/N]Frequency Response H1(Response,Excitation) - Input (Magnitude)Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100m
10
[Hz]
[(m/s)/N]
Time Domain
Impulse Response h1(Response,Excitation) - Input (Real Part)
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m-2k
-1k
0
1k
2k
[s]
[(m/s)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m-2k
-1k
0
1k
2k
[s]
[(m/s)/N/s]Inverse
FFT
Excitation
Response
Force
Motion
Input
OutputH ===
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Modal Analysis 21
Hammer Test on Free-free Beam
Frequency
Distanc
eAmplitude First
Mode
Beam
SecondModeThirdMode
ForceForceForceForceForceForceForceForceForceForceForce
Roving hammer method:
Response measured at one point
Excitation of the structure at a number of pointsby hammer with force transducer
FrequencyDomainView
Modes of structure identified
FRFs between excitation points and measurement point calculated
Press anywhereto advance animation
Force
Modal
Doma
in
View
Acceleration
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Modal Analysis 22
Measurement of FRF Matrix (SISO)
One row
One Roving Excitation
One Fixed Response (reference)
SISO
=
X1 H11 H12 H13 ...H1n F1X2 H21H22H23 ...H2n F2
X3 H31 H32 H33...H 3n F3: : :
Xn Hn1 Hn2 Hn3...Hnn Fn
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Modal Analysis 23
Measurement of FRF Matrix (SIMO)
More rows
One Roving Excitation
Multiple Fixed Responses (references)
SIMO
=
X1 H11 H12 H13 ...H1n F1X2 H21H22H23 ...H2n F2
X3 H31 H32H33...H 3n F3: : :
Xn Hn1 Hn2 Hn3...Hnn Fn
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Modal Analysis 24
FRFs between excitation point and measurement points calculated
Whitenoiseexcitation
Shaker Test on Free-free Beam
Frequency
Distan
ceAmplitude First
Mode SecondMode
ThirdMode
Beam
Acceleration
Press anywhereto advance animation
Modal
Doma
in
View
FrequencyDomainView Force
Shaker method:
Excitation of the structure at one pointby shaker with force transducer
Response measured at a number of points
Modes of structure identified
f (S S O)
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Modal Analysis 25
Measurement of FRF Matrix (Shaker SIMO)
One column Single Fixed Excitation (reference)
Single Roving Response SISO
or Multiple (Roving) Responses SIMO
Multiple-Output: Optimize data consistency
X1 H11H12 H13 ...H1n F1X2 H21H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3: : :
Xn Hn1Hn2 Hn3...Hnn Fn
=
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Modal Analysis 26
Why Multiple-Input and Multiple-Output ?
Multiple-Input: For large and/or complex structures moreshakers are required in order to:
get the excitation energy sufficiently distributed
and
avoid non-linear behaviour
Multiple-Output: Measure outputs at the same time inorder to optimize data consistency
i.e. MIMO
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Modal Analysis 27
Situations needing MIMO
One row or one column is not sufficient for determination ofall modes in following situations:
More modes at the same frequency (repeated roots),
e.g. symmetrical structures
Complex structures having local modes, i.e. referenceDOF with modal deflection for all modes is not available
In both cases more columns or more rows have to bemeasured - i.e. polyreference.
Solutions:
Impact Hammerexcitation with more response DOFs One shakermoved to different reference DOFs
MIMO
M f FRF M i (MIMO)
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Modal Analysis 28
Measurement of FRF Matrix (MIMO)
=
More columns Multiple Fixed Excitations (references)
Single Roving Response MISOor
Multiple (Roving) Responses MIMO
X1 H11H12 H13 ...H1n F1X2 H21H22H23 ...H2n F2X
3 H31H
32H
33...H 3n F
3: : :
Xn Hn1Hn2Hn3...Hnn Fn
( ) /
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Modal Analysis 29
Modal Analysis (classic): FRF = Response/Excitation
Output
Frequency Response H1(Response,Excitation) - Input (Magnitude)Frequency Response H1(Response,Excitation) - Input (Magnitude)
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100m
10
[Hz]
[(m/s)/N]
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100m
10
[Hz]
[(m/s)/N]
Autospectrum(Excitation) -Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100u
1m
10m
100m
1
[Hz]
[N] Autospectrum(Excitation) -Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100u
1m
10m
100m
1
[Hz]
[N]
Autospectrum(Response) -Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
1m
10m
100m
1
10
[Hz]
[m/s] Autospectrum(Response) -Input
Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
1m
10m
100m
1
10
[Hz]
[m/s]
Time(Excitation) - Input
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N] Time(Excitation) - Input
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N]
Time(Response) - Input
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s] Time(Response) - Input
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s]
FFT
FFT
Impulse Response h1(Response,Excitation) - Input (Real Part)
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-2k
-1k
0
1k
2k
[s]
[(m/s)/N/s]Impulse Response h1(Response,Excitation) - Input (Real Part)
Working : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-2k
-1k
0
1k
2k
[s]
[(m/s)/N/s]
Inverse
FFT
Input
Excitation
Response
Force
Vibration
Input
OutputH() = = =
FrequencyResponseFunction
Frequency Domain
ImpulseResponseFunction
Time Domain
Operational Modal Analysis (OMA): Response only!
Natural Excitation
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Modal Analysis 30
SDOF and MDOF Models
Different Modal AnalysisTechniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis PostProcessing
f(t)
x(t)
kc
m
= ++ + +
Th Et l Q ti i M d l
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Modal Analysis 31
The Eternal Question in Modal
F1a
F2
Impact Excitation
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Modal Analysis 32
Impact Excitation
H11() H12() H15()
Measuring one row of the FRF matrix bymoving impact position
LAN
# 1
# 2
# 3# 4
# 5
Accelerometer
ForceTransducer Impact
Hammer
Impact Excitation
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Modal Analysis 33
Impact Excitation
t
GAA(f)
Magnitude and pulse duration depends on: Weight of hammer
Hammer tip (steel, plastic or rubber)
Dynamic characteristics of surface
Velocity at impact
Frequency bandwidth inversely proportional to the pulse duration
a(t)
a(t)
ft
1
2
1
2
Weighting Functions for Impact Excitation
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Modal Analysis 34
Weighting Functions for Impact Excitation
How to select shift and lengthfor transient and exponentialwindows:
Leakage due to exponential timeweighting on response signal is welldefined and therefore correction of themeasured damping value is often possible
Transient weighting of the input signal
Criteria
Exponential weightingof the output signal
Compensation for Exponential Weighting
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Modal Analysis 35
Compensation for Exponential Weighting
With exponentialweighting of theoutput signal, themeasured time
constant will be tooshort and thecalculated decayconstant anddamping ratio
therefore too large shift Length =Record length, T
Original signal
Weighted signal
Window function
Time
1
b(t)
W
Correction of decay constant and damping ratio
Wm =
W
W
1
=
Wm
0
W
0
m
0
=
=
=
Correctvalue
Measuredvalue
Range of hammers
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Modal Analysis 36
3 lb HandSledge
Building andbridges
12 lb Sledge
Large shaftsand largermachine tools
Car framed
and machinetools
1 lb hammer
ComponentsGeneralPurpose, 0.3
lbHard-drives,circuit boards,turbine blades
Mini Hammer
ApplicationDescription
Range of hammers
Impact hammer excitation
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Modal Analysis 37
Impact hammer excitation
Advantages: Speed
No fixturing
No variable mass loading
Portable and highly suitable forfield work
relatively inexpensive
Disadvantages High crest factor means
possibility of driving structureinto non-linear behavior
High peak force needed forlarge structures meanspossibility of local damage!
Highly deterministic signalmeans no linear approximation
Conclusion
Best suited for field work
Useful for determining shakerand support locations
Shaker Excitation
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Modal Analysis 38
Shaker Excitation
H11()
H21()
H51()
Measuring one column of the FRF matrix bymoving response transducer
# 1
# 2
# 3# 4
# 5Accelerometer
ForceTransducer
VibrationExciter
PowerAmplifier
LAN
Attachment of Transducers and Shaker
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Modal Analysis 39
Attachment of Transducers and Shaker
Accelerometer mounting:
Stud
Cement
Wax
(Magnet)
Force Transducer and Shaker:
Stud
Stinger (Connection Rod)
Properties of Stinger
Axial Stiffness: High
Bending Stiffness: Low
Accelerometer
ShakerFa
ForceTransducer
Advantages of Stinger:
No Moment Excitation
No Rotational Inertia Loading Protection of Shaker
Protection of Transducer
Helps positioning of Shaker
Connection of Exciter and Structure
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Modal Analysis 40
Connection of Exciter and Structure
ExciterSlenderstinger
Measured structure
Accelerometer
Force Transducer
Tip mass, m
Shaker/Hammermass, M
Fs
Fm
Piezoelectricmaterial
Structure
M
MmFF ms
+=
XMFm &&=
X)Mm(Fs&&+=
Force and acceleration measurements unaffected by stinger compliance, but ...
Minor mass correction required to determine actual excitation
Shaker Reaction Force
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Modal Analysis 41
Shaker Reaction Force
Reaction byexternal support
Reaction byexciter inertia
Example of animproper
arrangement
Structure
Suspension
Exciter
Support
Structure
Suspension
Exciter
Suspension
Sine Excitation
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Modal Analysis 42
Sine Excitation
a(t)
Time
A
Crest factor 2RMSA ==
For study of non-linearities,
e.g. harmonic distortion
For broadband excitation: Sine wave swept slowly through
the frequency range of interest
Quasi-stationary condition
A(f1)
B(f1)
RMS
Swept Sine Excitation
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Modal Analysis 43
p
Advantages Low Crest Factor
High Signal/Noise ratio
Input force well controlled
Study of non-linearities
possibleDisadvantages
Very slow
No linear approximation ofnon-linear system
Random Excitation
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Modal Analysis 44
A(f1)Freq.
Time
a(t)
B(f1)GAA(f), N = 1 GAA(f), N = 10
Random variation of amplitude and phaseAveraging will give optimum linear
estimate in case of non-linearities
Freq.
SystemOutput
System
Input
Random Excitation
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Modal Analysis 45
Random signal:
Characterized by power spectral density (GAA) andamplitude probability density (p(a))
a(t)
p(a)
Can be band limited according to frequency range of interest
GAA(f) GAA(f)Baseband Zoom
Time
Signal not periodic in analysis time Leakage in spectral estimates
Frequencyrange
Frequencyrange
Freq. Freq.
Random Excitation
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Modal Analysis 46
Advantages Best linear approximation of system
Zoom Fair Crest Factor
Fair Signal/Noise ratio
Disadvantages Leakage
Averaging needed (slower)
Burst Random
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Modal Analysis 47
Characteristics of Burst Random signal :
Gives best linear approximation of nonlinear system
Works with zoom
a(t)
Time
Advantages Best linear approximation of system
No leakage (if rectangular time weighting can be used)
Relatively fast
Disadvantages Signal/noise and crest factor not optimum
Special time weighting might be required
Pseudo Random Excitation
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Modal Analysis 48
a(t)
Pseudo random signal:
Block of a random signal repeated every T
Time
A(f1)
B(f1)
T T T T
Time period equal to record length T
Line spectrum coinciding with analyzer lines
No averaging of non-linearities
Freq.
GAA(f), N = 1 GAA(f), N = 10
Freq.
SystemOutput
System Input
Pseudo Random Excitation
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Modal Analysis 49
T T T T
a(t)
Time
p(a)
Freq. range Freq. range
GAA(f) GAA(f)Baseband Zoom
Can be band limited according to frequency range of interest
Time period equal to T No leakage if Rectangular weighting is used
Pseudo random signal:
Characterized by power/RMS (GAA) and amplitude probability density (p(a))
Freq. Freq.
Pseudo Random Excitation
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Modal Analysis 50
Advantages No leakage
Fast Zoom
Fair crest factor
Fair Signal/Noise ratioDisadvantages
No linear approximation ofnon-linear system
Multisine (Chirp)
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Modal Analysis 51
For sine sweep repeated every time record, Tr
Tr
A special type of pseudo random signal wherethe crest factor has been minimized (< 2)
It has the advantages and disadvantages of
the normal pseudo random signal but with alower crest factor
Additional Advantages:
Ideal shape of spectrum: The spectrum is a flatmagnitude spectrum, and the phase spectrum is smooth
Applications:
Measurement of structures with non-linear behaviour
Time
Periodic Random
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Modal Analysis 52
Pseudo-random signalchanging with time:
Analysed time data:(steady-state response)
transient response
steady-state response
A B C
A A A B B B C CC
T T T
A combined random and pseudo-random signal givingan excitation signal featuring:
No leakage in analysis
Best linear approximation of system
Disadvantage:
The test time is longer than the test timeusing pseudo-random or random signal
Periodic Pulse
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Modal Analysis 53
Leakage can be avoided using rectangular time weighting
Transient and exponential time weighting can be used to increase
Signal/Noise ratio Gating of reflections with transient time weighting
Effects of non-linearities are not averaged out
The signal has a high crest factor
Rectangular, Hanning, or Gaussian pulse with user definable t repeated with a user definable interval, T
The line spectrum for a
Rectangular pulse has a
shaped envelope curve
sin xx
t T Time
1/ t Frequency
Special case of pseudo random signal
Periodic Pulse
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Modal Analysis 54
Advantages
Fast
No leakage(Only with rectangular weighting)
Gating of reflections
(Transient time weighting) Excitation spectrum follows
frequency span in baseband
Easy to implement
Disadvantages
No linear approximation ofnon-linear system
High Crest Factor
High peak level might excite
non-linearities No Zoom
Special time weighting mightbe required to increase
Signal/Noise Ratio . This canalso introduce leakage
Guidelines for Choice of Excitation Technique
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Modal Analysis 55
Swept sine excitation For study of non-linearities:
Random excitation For slightly non-linear system:
Pseudo random excitation For perfectly linear system:
Impact excitation For field measurements:
Random impact excitation For high resolution field measurements:
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Modal Analysis 56
SDOF and MDOF Models
Different Modal Analysis
Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post
Processing
f(t)
x(t)
kc
m
= ++ + +
Garbage In = Garbage Out!
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Modal Analysis 57
A state-of-the Art Assessment of Mobility Measurement Techniques
Result for the Mid Range Structure (30 - 3000 Hz)
D.J. Ewins and J. GriffinFeb. 1981
Transfer MobilityCentral Decade
Transfer MobilityExpanded
Frequency Frequency
Plan Your Test Before Hand!
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Modal Analysis 58
1. Select Appropriate Excitation Hammer, Shaker, or OMA?
2. Setup FFT Analyzer correctly Frequency Range, Resolution, Averaging, Windowing
Remember: FFT Analyzer is a BLOCK ANALYZER!
3. Good Distribution of Measurement Points Ensure enough points are measured to see all modes of interest
Beware of spatial aliasing
4. Physical Setup Accelerometer mounting is CRITICAL!
Uni-axial vs. Triaxial Make sure DOF orientation is correct
Mount device under test...mounting will affect measurement!
Calibrate system
Where Should Excitation Be Applied?
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Modal Analysis 59
Driving PointMeasurement
TransferMeasurement
i = j
i j
i
j
j
i
"j"Excitation"i"sponseRe
FXH
j
iij ==
12
2121111 FHFHX +=
=
2
1
2221
1211
2
1
F
F
HH
HH
X
X
}{ [ ]{ }FHX =
2221212 FHFHX +=
Check of Driving Point Measurement
I [H ]
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Modal Analysis 60
All peaks in
)f(F
)f(XImand
)f(F
)f(XRe,
)f(F
)f(XIm
&&&
An anti-resonance in Mag[Hij]
must be found between everypair of resonances
Phase fluctuations must bewithin 180
Im[Hij]
Mag[Hij]
Phase [Hij]
Driving Point (DP) Measurement
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Modal Analysis 61
The quality of the DP-measurement is very important, as the
DP-residues are used for the scaling of the Modal Model
DP- Considerations:
Residues for all modesmust be estimatedaccurately from a singlemeasurement
DP- Problems:
Highest modal coupling, as allmodes are in phase
Highest residual effect fromrigid body modes
m Aij
Re Aij
Log MagAij
=
Without rigid
body modes
Measured
Tests for Validity of Data: Coherence
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Modal Analysis 62
Coherence)f(G)f(G
)f(G)f(XXFF
2
FX2 =
Measures how much energy put in to thesystem caused the response
The closer to 1 the more coherent
Less than 0.75 is bordering on poor coherence
Reasons for Low Coherence
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Modal Analysis 63
Difficult measurements:
Noise in measured output signal
Noise in measured input signal
Other inputs not correlated with measured input signal
Bad measurements:
Leakage Time varying systems
Non-linearities of system
DOF-jitter
Propagation time not compensated for
Tests for Validity of Data: Linearity
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Modal Analysis 64
Linearity X1 = HF1X2 = HF2
X1+X2 = H(F1 + F2)aX1 = H(a F1)
More force going in to the system will equate to
F() X()
more response coming out
Since FRF is a ratio the magnitude should bethe same regardless of input force
H()
Tips and Tricks for Best Results
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Modal Analysis 65
Verify measurement chain integrity prior to test:
Transducer calibration
Mass Ratio calibration
Verify suitability of input and output transducers:
Operating ranges (frequency, dynamic range, phase response)
Mass loading of accelerometers
Accelerometer mounting
Sensitivity to environmental effects
Stability
Verify suitability of test set-up:
Transducer positioning and alignment
Pre-test: rattling, boundary conditions, rigid body modes, signal-to-noise ratio,linear approximation, excitation signal, repeated roots, Maxwell reciprocity,
force measurement, exciter-input transducer-stinger-structure connection
Quality FRF measurements are the foundation ofexperimental modal analysis!
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Modal Analysis 66
SDOF and MDOF Models
Different Modal Analysis
Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post
Processing
f(t)
x(t)
kc
m
= ++ + +
From Testing to Analysis
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Modal Analysis 67
MeasuredFRF
Curve Fitting
(Pattern Recognition)
Modal Analysis
)f(H
Frequency
)f(H
Frequency
From Testing to Analysis
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Modal Analysis 68
Modal Analysis
)f(H
Frequency
f(t)
x(t)
kc
m
f(t)
x(t)
kc
m
f(t)
x(t)
kc
mSDOF Models
Mode Characterizations
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Modal Analysis 69
All Modes Can Be Characterized By:
1. Resonant Frequency
2. Modal Damping
3. Mode Shape
Modal Analysis Step by Step Process
1. Visually Inspect Data
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Modal Analysis 70
y p
Look for obvious modes in FRF InspectALL FRFssometimes modes will show up in one
FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis Use more than 1 curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?
Modal Analysis Inspect Data
1. Visually Inspect Data
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Modal Analysis 71
Look for obvious modes in FRF InspectALL FRFssometimes modes will show up in one
FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis Use more than 1 curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?
Modal Analysis Curve Fitting
1. Visually Inspect Data
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Modal Analysis 72
Look for obvious modes in FRF InspectALL FRFssometimes modes will show up in one
FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis Use more than 1 curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?
How Does Curve Fitting Work?
Curve Fitting is the process of estimating the Modal
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Modal Analysis 73
Curve Fitting is the process of estimating the ModalParameters from the measurements
|H|
2
R/
d
0
-180
Phase
Frequency
Find the resonant
frequency Frequency where smallexcitation causes alarge response
Find the damping What is the Q of the
peak?
Find the residue Essentially the area
under the curve
Residues are Directly Related to Mode Shapes!
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Modal Analysis 74
*
*
r
ijr
r r
ijr
rijrij
pj
R
pj
RH)(H
+
==
Residues express thestrength of a mode for eachmeasured FRF
Therefore they are relatedto mode shape at eachmeasured point!
Frequency
Dista
nce
Amplitude FirstMode
Beam
SecondMode
ForceForceForceForceForceForceForceForceForceForceForceForce
Acceleration
ThirdMode
SDOF vs. MDOF Curve Fitters
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Modal Analysis 75
Use SDOF methods on LIGHTLY COUPLED modes Use MDOF methods on HEAVILY COUPLED modes
You can combine SDOF and MDOF techniques!
SDOF
(light coupling)
|H|
MDOF
(heavy coupling)
Local vs. Global Curve Fitting
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Modal Analysis 76
Local means that resonances, damping, and residues arecalculated for each FRF firstthen combined for curvefitting
Global means that resonances, damping, and residues are
calculated across all FRFs
|H|
Modal Analysis Analyse Results
1. Visually Inspect Data
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Modal Analysis 77
Look for obvious modes in FRF InspectALL FRFssometimes modes will show up in one
FRF but not another (nodes)
Use Imaginary part and coherence for verification
Sum magnitudes of all measurements for clues2. Select Curve Fitter
Lightly coupled modes: SDOF techniques
Heavily coupled modes: MDOF techniques
Stable measurements: Global technique
Unstable measurements: Local technique
MIMO measurement: Poly reference techniques
3. Analysis
Use more than one curve fitter to see if they agree
Pay attention to Residue calculations
Do mode shapes make sense?
Which Curve Fitter Should Be Used?
Frequency Response Function
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Modal Analysis 78
q y p
Hij ()
Real Imaginary
Nyquist Log Magnitude
Which Curve Fitter Should Be Used?
Frequency Response Function
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Modal Analysis 79
Real Imaginary
Nyquist Log Magnitude
Hij ()
Modal Analysis and Beyond
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Modal Analysis 80
Hardware ModificationResonance Specification
Simulate RealWorld Response
Dynamic Modelbased on
Modal Parameters
ExperimentalModal Analysis
Structural
Modification
Response
Simulation
F
Conclusion
All Physical Structures can be characterized by the simple
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Modal Analysis 81
SDOF model
Frequency Response Functions are the best way tomeasure resonances
There are three modal techniques available today:Hammer Testing, Shaker Testing, and Operational Modal
Planning and proper setup before you test can save timeand effortand ensure accuracy while minimizingerroneous results
There are many curve fitting techniques available, try touse the one that best fits your application
Literature for Further Reading
Structural Testing Part 1: Mechanical Mobility Measurements
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Modal Analysis 82
Brel & Kjr Primer
Structural Testing Part 2: Modal Analysis and Simulation
Brel & Kjr Primer
Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J.Ewin
Research Studies Press Ltd.
Dual Channel FFT Analysis (Part 1)
Brel & Kjr Technical Review # 1 1984
Dual Channel FFT Analysis (Part 1)
Brel & Kjr Technical Review # 2 1984
Appendix: Damping Parameters
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Modal Analysis 83
3 dB bandwidth=
22fdB3
Loss factor 0
dB3
0
db3
f
f
Q
1
=
==
Damping ratio
0
Bd3
0
dB3
2f2
f
2
=
=
=
Decay constant
Quality factordB3
0
dB3
0
f
fQ
=
=
2f dB3dB30
===
= 2dB3,
Appendix: Damping Parameters
t -t
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Modal Analysis 84
( )tsineR2)t(h d= , where the Decay constant is given by e
The Envelope is given by magnitude of analytic h(t):
The time constant, , isdetermined by the time it
takes for the amplitude to
decay a factor of e = 2,72
or
10 log (e2) = 8.7 dB
)t(h~
)t(h)t(h 22 +=
Decay constant
Time constant :
Damping ratio
Loss factor
Quality
= 1
=1
=
=00 f2
1
==
0f
12
=
= 0f1
Q
h(t)
Time
t
e
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