Modal Analysis 1
ExperimentalModal Analysis
Copyright 2003Brel & Kjr Sound & Vibration Measurement A/SAll Rights Reserved
Modal Analysis 2
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing
f(t)
x(t)
kc
m
= ++ + +
Modal Analysis 3
Simplest Form of Vibrating System
D
d = D sinnt
m
k
T
Time
Displacement
Frequency1T
Period, Tn in [sec]
Frequency, fn= in [Hz = 1/sec] 1Tn
Displacement
kmn= 2 fn =
Modal Analysis 4
Mass and Spring
time
m1
m
Increasing mass reduces frequency
1n mm
k2 +== nf
Modal Analysis 5
Mass, Spring and Damper
Increasing damping reduces the amplitude
time
m
k c1 + c2
Modal Analysis 6
)()()()( tftKxtxCtxM =++ &&&M = mass (force/acc.)C = damping (force/vel.)K = stiffness (force/disp.)
=)t(x&&=)t(x&=)t(x
Acceleration VectorVelocity VectorDisplacement Vector
=)t(f Applied force Vector
Basic SDOF Model
f(t)x(t)
kc
m
Modal Analysis 7
SDOF Models Time and Frequency Domain
kcjmFXH ++==
2 1)()()(
H()F() X()
|H()|1k
H()
0 90
180
0 = k/m
12m
1c
)()()()( tkxtxctxmtf ++= &&&
f(t)x(t)
kc
m
Modal Analysis 8
Modal Matrix
Modal Model(Freq. Domain)
( )( )( )
( )
( ) ( ) ( )( )( )
( ) ( )
( )
=
3
1
23
22
12111
3
2
1
F
HH
HH
HHH
X
XXX
nnn
n
n
X1
X2X3
X4
F3H21
H22
Modal Analysis 9
MDOF Model
0
-90
-180
2 1
1+2
1 2
1+2
Frequency
FrequencyPhase
Magnitude
m
d1+ d2
d1
dF
Modal Analysis 10
Why Bother with Modal Models?
Physical Coordinates = CHAOS Modal Space = Simplicity
q1
11
20112 2
q21
20222
q31
20332
3
Bearing
Rotor
Bearing
Foundation
Modal Analysis 11
Definition of Frequency Response Function
F X
f f f
H(f) is the system Frequency Response FunctionF(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
Modal Analysis 12
Benefits of Frequency Response Function
F(f) H(f) X(f)
z Frequency Response Functions are properties of linear dynamic systems
z They are independent of the Excitation Function
z Excitation can be a Periodic, Random or Transient function of time
z The test result obtained with one type of excitation can be used for predicting the response of the system to any other type of excitation
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
Modal Analysis 14
Alternative Estimators
)()()( fFfXfH =
)()()(1 fGfGfH
FF
FX=
)()()(2 fGfGfH
XF
XX=
213 )( HHGG
GGfH
FX
FX
FF
XX ==
2
1
22 *)( H
HGG
GG
GGG
fXX
FX
FF
FX
XXFF
FX ===
X(f)F(f) H(f)
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 H3Input noise - Best -
Output noise Best - -
Input + output noise - - Best
Peaks (leakage) - Best -
Valleys (leakage) Best - -
Accuracy
)()()(1 fGfGfH
FF
FX= )()()(2 fGfGfH
XF
XX=FX
FX
FF
XX
GG
GGfH =)(3
Modal Analysis 16
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing
f(t)
x(t)
kc
m
= ++ + +
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
Modal Analysis 18
Different Types of Modal Analysis (Pros)
z Hammer Testing Quick and easy Typically Inexpensive Can perform poor man modal as well as full modal
z Shaker Testing More repeatable than hammer testing Many types of input available Can be used for MIMO analysis
z Operational Modal Analysis No need for special boundary conditions Measure in-situ Use natural excitation Can perform other tests while taking OMA data
Modal Analysis 19
Different Types of Modal Analysis (Cons)
z 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
z Shaker Testing More difficult test setup (stingers, exciter, etc.) Usually more equipment and channels needed Skilled operator(s) needed
z Operational Modal Analysis Unscaled modal model Excitation assumed to cover frequency range of interest Long time histories sometimes required Computationally intensive
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) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N]
FFT
FFTFrequency Domain
Autospectrum(Excitation) - InputWorking : 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) - InputWorking : 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) - InputWorking : 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) - InputWorking : 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]InverseFFT
ExcitationResponse
ForceMotion
InputOutputH ===
Modal Analysis 21
Hammer Test on Free-free Beam
Frequency
Distan
ceAmplitude First
Mode
Beam
SecondMode Third
Mode
ForceForceForceForceForceForceForceForceForceForceForce
Roving hammer method:z Response measured at one point z Excitation of the structure at a number of points
by hammer with force transducer
Frequency Domain View
z Modes of structure identifiedz FRFs between excitation points and measurement point calculated
Press anywhereto advance animation
ForceMo
dal
Domain
View
Acceleration
Modal Analysis 22
Measurement of FRF Matrix (SISO)
One rowz One Roving Excitationz One Fixed Response (reference)
SISO
=
X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn
Modal Analysis 23
Measurement of FRF Matrix (SIMO)
More rowsz One Roving Excitationz Multiple Fixed Responses (references)
SIMO
=
X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn
Modal Analysis 24
z FRFs between excitation point and measurement points calculated
Whitenoiseexcitation
Shaker Test on Free-free Beam
Frequency
Distan
ceAmplitude First
Mode SecondMode
ThirdMode
BeamAcceleration
Press anywhereto advance animation
Modal
Domain
View
Frequency Domain View Force
Shaker method:z Excitation of the structure at one point
by shaker with force transducerz Response measured at a number of points
z Modes of structure identified
Modal Analysis 25
Measurement of FRF Matrix (Shaker SIMO)
One columnz Single Fixed Excitation (reference)
z Single Roving Response SISOor
z Multiple (Roving) Responses SIMOMultiple-Output: Optimize data consistency
X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn
=
Modal Analysis 26
Why Multiple-Input and Multiple-Output ?z Multiple-Input: For large and/or complex structures more
shakers are required in order to: get the excitation energy sufficiently distributedand avoid non-linear behaviour
z Multiple-Output: Measure outputs at the same time in order to optimize data consistency
i.e. MIMO
Modal Analysis 27
Situations needing MIMO
z One row or one column is not sufficient for determination of all modes in following situations:
More modes at the same frequency (repeated roots), e.g. symmetrical structures
Complex structures having local modes, i.e. reference DOF with modal deflection for all modes is not available
In both cases more columns or more rows have to be measured - i.e. polyreference.
Solutions: Impact Hammer excitation with more response DOFs One shaker moved to different reference DOFs MIMO
Modal Analysis 28
Measurement of FRF Matrix (MIMO)
=
More columnsz Multiple Fixed Excitations (references)
z Single Roving Response MISOor
z Multiple (Roving) Responses MIMO
X1 H11 H12 H13 ...H1n F1X2 H21 H22 H23 ...H2n F2X3 H31 H32 H33...H 3n F3: : :Xn Hn1 Hn2 Hn3...Hnn Fn
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) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N] Time(Excitation) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-200
-100
0
100
200
[s]
[N]
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]
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]
InverseFFT
Input
ExcitationResponse
ForceVibration
InputOutput
H() = = =
FrequencyResponseFunction
Frequency Domain
ImpulseResponseFunction
Time Domain
Operational Modal Analysis (OMA): Response only!
Natural Excitation
Modal Analysis 30
SDOF and MDOF Models
Different Modal Analysis Techniques
Exciting a Structure
Measuring Data Correctly
Modal Analysis Post Processing
f(t)
x(t)
kc
m
= ++ + +
Modal Analysis 31
The Eternal Question in Modal
F1a
F2
Modal Analysis 32
Impact Excitation
H11() H12() H15()
Measuring one row of the FRF matrix by moving impact position
LAN
# 1# 2
# 3# 4
# 5Accelerometer
ForceTransducer Impact
Hammer
Modal Analysis 33
Impact Excitation
t
GAA(f)
z Magnitude and pulse duration depends on: Weight of hammer Hammer tip (steel, plastic or rubber) Dynamic characteristics of surface Velocity at impact
z Frequency bandwidth inversely proportional to the pulse durationa(t)
a(t)
ft
12
12
Modal Analysis 34
Weighting Functions for Impact Excitation
z How to select shift and length for transient and exponential windows:
z Leakage due to exponential time weighting on response signal is well defined and therefore correction of the measured damping value is often possible
Transient weighting of the input signal
Criteria
Exponential weighting of the output signal
Modal Analysis 35
Compensation for Exponential Weighting
With exponential weighting of the output signal, the measured time constant will be too short and the calculated decay constant and damping ratio therefore too large shift Length =
Record length, T
Original signal
Weighted signal
Window function
Time
1b(t)
W
Correction of decay constant and damping ratio :
Wm =
WW
1=
Wm0
W
0
m
0
=
==
Correctvalue
Measuredvalue
Modal Analysis 36
3 lb Hand Sledge
Building and bridges
12 lb Sledge
Large shafts and larger machine toolsCar framed and machine tools
1 lb hammer
ComponentsGeneral Purpose, 0.3 lb
Hard-drives, circuit boards, turbine blades
Mini Hammer
ApplicationDescription
Range of hammers
Modal Analysis 37
Impact hammer excitation
z Advantages: Speed No fixturing No variable mass loading Portable and highly suitable for
field work relatively inexpensive
z Disadvantages High crest factor means
possibility of driving structure into non-linear behavior
High peak force needed for large structures means possibility of local damage!
Highly deterministic signal means no linear approximation
z Conclusion Best suited for field work Useful for determining shaker and support locations
Modal Analysis 38
Shaker Excitation
H11() H21() H51()
Measuring one column of the FRF matrix by moving response transducer
# 1# 2
# 3# 4
# 5Accelerometer
ForceTransducer
VibrationExciter PowerAmplifier
LAN
Modal Analysis 39
Attachment of Transducers and Shaker
Accelerometer mounting:z Studz Cementz Waxz (Magnet)
Force Transducer and Shaker:z Studz Stinger (Connection Rod)
Properties of StingerAxial Stiffness: HighBending Stiffness: Low
Accelerometer
ShakerFa
ForceTransducer
Advantages of Stinger: z No Moment Excitationz No Rotational Inertia Loadingz Protection of Shakerz Protection of Transducerz Helps positioning of Shaker
Modal Analysis 40
Connection of Exciter and Structure
ExciterSlenderstinger
Measured structure
Accelerometer
Force Transducer
Tip mass, mShaker/Hammer
mass, M
FsFm
Piezoelectricmaterial
Structure
MMmFF ms
+=
XMFm &&=X)Mm(Fs &&+=
Force and acceleration measurements unaffected by stinger compliance, but ...Minor mass correction required to determine actual excitation
Modal Analysis 41
Shaker Reaction Force
Reaction byexternal support
Reaction byexciter inertia
Example of animproper
arrangementStructure
Suspension
ExciterSupport
StructureSuspension
ExciterSuspension
Modal Analysis 42
Sine Excitationa(t)
Time
A
Crest factor 2RMSA ==
z For study of non-linearities, e.g. harmonic distortion
z For broadband excitation: Sine wave swept slowly through
the frequency range of interest Quasi-stationary condition
A(f1)
B(f1)
RMS
Modal Analysis 43
Swept Sine Excitation
Advantagesz Low Crest Factorz High Signal/Noise ratioz Input force well controlledz Study of non-linearities
possible
Disadvantagesz Very slowz No linear approximation of
non-linear system
Modal Analysis 44
Random Excitation
A(f1)Freq.
Time
a(t)
B(f1)GAA(f), N = 1 GAA(f), N = 10
Random variation of amplitude and phase Averaging will give optimum linear
estimate in case of non-linearities
Freq.
SystemOutput
SystemInput
Modal Analysis 45
Random Excitationz Random signal:
Characterized by power spectral density (GAA) and amplitude probability density (p(a))
a(t)
p(a)
z Can be band limited according to frequency range of interestGAA(f) GAA(f)Baseband Zoom
Time
z Signal not periodic in analysis time Leakage in spectral estimatesFrequency
rangeFrequency
range
Freq. Freq.
Modal Analysis 46
Random Excitation
Advantagesz Best linear approximation of systemz Zoomz Fair Crest Factorz Fair Signal/Noise ratio
Disadvantagesz Leakagez Averaging needed (slower)
Modal Analysis 47
Burst Randomz Characteristics of Burst Random signal :
Gives best linear approximation of nonlinear system Works with zoom
a(t)
Time
Advantagesz Best linear approximation of systemz No leakage (if rectangular time weighting can be used)z Relatively fast
Disadvantagesz Signal/noise and crest factor not optimumz Special time weighting might be required
Modal Analysis 48
Pseudo Random Excitation
a(t)
z Pseudo random signal: Block of a random signal repeated every T
Time
A(f1)
B(f1)
T T T T
z 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
Modal Analysis 49
Pseudo Random Excitation
T T T T
a(t)
Timep(a)
Freq. range Freq. range
GAA(f) GAA(f)Baseband Zoom
z Can be band limited according to frequency range of interest
Time period equal to T No leakage if Rectangular weighting is used
z Pseudo random signal: Characterized by power/RMS (GAA) and amplitude probability density (p(a))
Freq. Freq.
Modal Analysis 50
Pseudo Random Excitation
Advantagesz No leakagez Fastz Zoomz Fair crest factorz Fair Signal/Noise ratio
Disadvantagesz No linear approximation of
non-linear system
Modal Analysis 51
Multisine (Chirp)For sine sweep repeated every time record, Tr
TrA special type of pseudo random signal where the crest factor has been minimized (< 2)
It has the advantages and disadvantages of the normal pseudo random signal but with a lower crest factor
Additional Advantages: Ideal shape of spectrum: The spectrum is a flat
magnitude spectrum, and the phase spectrum is smoothApplications:
Measurement of structures with non-linear behaviour
Time
Modal Analysis 52
Pseudo-random signal changing with time:
Analysed time data:(steady-state response)
transient responsesteady-state response
A B C
A A A B B B C CC
T T T
Periodic RandomA combined random and pseudo-random signal giving an excitation signal featuring:
No leakage in analysis Best linear approximation of system
Disadvantage: z The test time is longer than the test time
using pseudo-random or random signal
Modal Analysis 53
Periodic Pulse
z Leakage can be avoided using rectangular time weightingz Transient and exponential time weighting can be used to increase
Signal/Noise ratioz Gating of reflections with transient time weightingz Effects of non-linearities are not averaged outz 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
Modal Analysis 54
Periodic Pulse
Advantages
z Fastz No leakage
(Only with rectangular weighting)
z Gating of reflections(Transient time weighting)
z Excitation spectrum follows frequency span in baseband
z Easy to implement
Disadvantages
z No linear approximation of non-linear system
z High Crest Factorz High peak level might excite
non-linearitiesz No Zoomz Special time weighting might
be required to increase Signal/Noise Ratio . This can also introduce leakage
Modal Analysis 55
Guidelines for Choice of Excitation Technique
Swept sine excitationz For study of non-linearities:
Random excitationz For slightly non-linear system:
Pseudo random excitationz For perfectly linear system:
Impact excitationz For field measurements:
Random impact excitationz For high resolution field measurements:
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
= ++ + +
Modal Analysis 57
Garbage In = Garbage Out!A state-of-the Art Assessment of Mobility Measurement Techniques
Result for the Mid Range Structure (30 - 3000 Hz) D.J. Ewins and J. Griffin
Feb. 1981
Transfer MobilityCentral Decade
Transfer MobilityExpanded
Frequency Frequency
Modal Analysis 58
Plan Your Test Before Hand!
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
Modal Analysis 59
Where Should Excitation Be Applied?
Driving Point Measurement
Transfer Measurement
i = j
i j
i
j
j
i
"j"Excitation"i"sponseRe
FXH
j
iij ==
12
2121111 FHFHX +=
=
2
1
2221
1211
2
1
FF
HHHH
XX
}{ [ ]{ }FHX =
2221212 FHFHX +=
Modal Analysis 60
Check of Driving Point Measurement
z All peaks in
)f(F)f(XImand
)f(F)f(XRe,
)f(F)f(XIm
&&&
z An anti-resonance in Mag [Hij]must be found between every pair of resonances
z Phase fluctuations must be within 180
Im [Hij]
Mag [Hij]
Phase [Hij]
Modal Analysis 61
Driving Point (DP) MeasurementThe 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 modes
must be estimated accurately from a single measurement
DP- Problems: Highest modal coupling, as all
modes are in phase Highest residual effect from
rigid body modes
m Aij
Re Aij
Log MagAij
=
Without rigidbody modes
Measured
Modal Analysis 62
Tests for Validity of Data: Coherence
Coherence)f(G)f(G
)f(G)f(
XXFF
2FX2 =
Measures how much energy put in to the system caused the response
The closer to 1 the more coherent Less than 0.75 is bordering on poor coherence
Modal Analysis 63
Reasons for Low Coherence
Difficult measurements:z Noise in measured output signalz Noise in measured input signalz Other inputs not correlated with measured input signal
Bad measurements:z Leakagez Time varying systemsz Non-linearities of systemz DOF-jitterz Propagation time not compensated for
Modal Analysis 64
Tests for Validity of Data: Linearity
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 be
the same regardless of input force
H()
Modal Analysis 65
Tips and Tricks for Best Resultsz Verify measurement chain integrity prior to test:
Transducer calibration Mass Ratio calibration
z 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
z 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 of experimental modal analysis!
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
= ++ + +
Modal Analysis 67
From Testing to Analysis
MeasuredFRF
Curve Fitting(Pattern Recognition)
Modal Analysis
)f(H
Frequency
)f(H
Frequency
Modal Analysis 68
From Testing to Analysis
Modal Analysis
)f(H
Frequency
f(t)x(t)
kc
m
f(t)x(t)
kc
m
f(t)x(t)
kc
mSDOF Models
Modal Analysis 69
Mode Characterizations
All Modes Can Be Characterized By:1. Resonant Frequency
2. Modal Damping
3. Mode Shape
Modal Analysis 70
Modal Analysis Step by Step Process1. Visually Inspect Data
Look for obvious modes in FRF Inspect ALL 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 clues
2. 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 71
Modal Analysis Inspect Data1. Visually Inspect Data
Look for obvious modes in FRF Inspect ALL 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 clues
2. 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 72
Modal Analysis Curve Fitting1. Visually Inspect Data
Look for obvious modes in FRF Inspect ALL 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 clues
2. 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 73
How Does Curve Fitting Work?z Curve Fitting is the process of estimating the Modal
Parameters from the measurements
|H|
2
R/
d0
-180Phase
Frequency
z Find the resonant frequency
Frequency where small excitation causes a large response
z Find the damping What is the Q of the
peak?
z Find the residue Essentially the area
under the curve
Modal Analysis 74
*
*
r
ijr
r r
ijr
rijrij pj
Rpj
RH)(H +==
Residues are Directly Related to Mode Shapes!
z Residues express the strength of a mode for each measured FRF
z Therefore they are related to mode shape at each measured point!
Frequency
Distan
ceAmplitude First
Mode
Beam
SecondMode
ForceForceForceForceForceForceForceForceForceForceForceForce
Acceleration
ThirdMode
Modal Analysis 75
SDOF vs. MDOF Curve Fitters
z Use SDOF methods on LIGHTLY COUPLED modes
z Use MDOF methods on HEAVILY COUPLED modes
z You can combine SDOF and MDOF techniques!
SDOF(light coupling)
|H|
MDOF(heavy coupling)
Modal Analysis 76
Local vs. Global Curve Fitting
z Local means that resonances, damping, and residues are calculated for each FRF firstthen combined for curve fitting
z Global means that resonances, damping, and residues are calculated across all FRFs
|H|
Modal Analysis 77
Modal Analysis Analyse Results1. Visually Inspect Data
Look for obvious modes in FRF Inspect ALL 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 clues
2. 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?
Modal Analysis 78
Which Curve Fitter Should Be Used?Frequency Response Function
Hij ()
Real Imaginary
Nyquist Log Magnitude
Modal Analysis 79
Which Curve Fitter Should Be Used?
Real Imaginary
Nyquist Log Magnitude
Frequency Response FunctionHij ()
Modal Analysis 80
Modal Analysis and Beyond
Hardware ModificationResonance Specification
Simulate RealWorld Response
Dynamic Modelbased on
Modal Parameters
ExperimentalModal Analysis
StructuralModification
ResponseSimulation
F
Modal Analysis 81
Conclusionz All Physical Structures can be characterized by the simple
SDOF model
z Frequency Response Functions are the best way to measure resonances
z There are three modal techniques available today: Hammer Testing, Shaker Testing, and Operational Modal
z Planning and proper setup before you test can save time and effortand ensure accuracy while minimizing erroneous results
z There are many curve fitting techniques available, try to use the one that best fits your application
Modal Analysis 82
Literature for Further Readingz Structural Testing Part 1: Mechanical Mobility Measurements
Brel & Kjr Primer
z Structural Testing Part 2: Modal Analysis and SimulationBrel & Kjr Primer
z Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J. EwinResearch Studies Press Ltd.
z Dual Channel FFT Analysis (Part 1)Brel & Kjr Technical Review # 1 1984
z Dual Channel FFT Analysis (Part 1)Brel & Kjr Technical Review # 2 1984
Modal Analysis 83
Appendix: Damping Parameters
3 dB bandwidth =
22f dB3
Loss factor0
dB3
0
db3
ff
Q1
===
Damping ratio0
Bd3
0
dB3
2f2f
2 ===
Decay constant
Quality factordB3
0
dB3
0
ffQ
==
2f dB3dB30
===
= 2dB3,
Modal Analysis 84
Appendix: Damping Parameters
( )tsineR2)t(h dt = , where the Decay constant is given by e-tThe Envelope is given by magnitude of analytic h(t):
The time constant, , is determined by the time it takes for the amplitude to decay a factor of e = 2,72or 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 f21
== 0f12
== 0f1Q
h(t)
Time
te
ExperimentalModal AnalysisSimplest Form of Vibrating SystemMass and SpringMass, Spring and DamperBasic SDOF Model SDOF Models Time and Frequency DomainModal MatrixMDOF ModelWhy Bother with Modal Models?Definition of Frequency Response FunctionBenefits of Frequency Response FunctionDifferent Forms of an FRFAlternative EstimatorsWhich FRF Estimator Should You Use?Three Types of Modal AnalysisDifferent Types of Modal Analysis (Pros)Different Types of Modal Analysis (Cons)Hammer Test on Free-free BeamMeasurement of FRF Matrix (SISO)Measurement of FRF Matrix (SIMO)Shaker Test on Free-free BeamMeasurement of FRF Matrix (Shaker SIMO)Why Multiple-Input and Multiple-Output ?Situations needing MIMOMeasurement of FRF Matrix (MIMO)The Eternal Question in ModalImpact ExcitationImpact ExcitationWeighting Functions for Impact ExcitationCompensation for Exponential WeightingRange of hammersImpact hammer excitationShaker ExcitationAttachment of Transducers and ShakerConnection of Exciter and StructureShaker Reaction ForceSine ExcitationSwept Sine ExcitationRandom ExcitationRandom ExcitationRandom ExcitationBurst RandomPseudo Random ExcitationPseudo Random ExcitationPseudo Random ExcitationMultisine (Chirp)Periodic RandomPeriodic PulsePeriodic PulseGuidelines for Choice of Excitation TechniqueGarbage In = Garbage Out!Plan Your Test Before Hand!Where Should Excitation Be Applied?Check of Driving Point MeasurementDriving Point (DP) MeasurementTests for Validity of Data: CoherenceReasons for Low CoherenceTests for Validity of Data: LinearityTips and Tricks for Best ResultsFrom Testing to AnalysisFrom Testing to AnalysisMode CharacterizationsModal Analysis Step by Step ProcessModal Analysis Inspect DataModal Analysis Curve FittingHow Does Curve Fitting Work?Residues are Directly Related to Mode Shapes!SDOF vs. MDOF Curve FittersLocal vs. Global Curve FittingModal Analysis Analyse ResultsWhich Curve Fitter Should Be Used?Which Curve Fitter Should Be Used?Modal Analysis and BeyondConclusionLiterature for Further ReadingAppendix: Damping ParametersAppendix: Damping Parameters
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