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Modal Analysis 1
ExperimentalModal Analysis
Copyright© 2003Brüel & Kjær 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 sinωnt
m
k
T
Time
Displacement
Frequency1T
Period, Tn in [sec]
Frequency, fn= in [Hz = 1/sec] 1Tn
Displacement
kmωn= 2 π fn =
Modal Analysis 4
Mass and Spring
time
m1
m
Increasing mass reduces frequency
1n mm
k2+
=π=ω 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
++−==
ωωωωω 2
1)()()(
H(ω)F(ω) X(ω)
ω
|H(ω)|
1k
ω
∠ H(ω)0º
– 90º– 180º
ω0 = √ k/m
1ω2m
1ωc
)()()()( tkxtxctxmtf ++= &&&
f(t)x(t)
kc
m
Modal Analysis 8
Modal Matrix
Modal Model(Freq. Domain)
( )( )( )
( )
( ) ( ) ( )( )( )
( ) ( )
( )
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⋅⋅⋅
⋅⋅
⎥⎥⎥⎥⎥⎥
⎦
⎤
⎢⎢⎢⎢⎢⎢
⎣
⎡
⋅⋅⋅⋅⋅⋅⋅⋅⋅
⋅⋅
=
⎪⎪⎪
⎭
⎪⎪⎪
⎬
⎫
⎪⎪⎪
⎩
⎪⎪⎪
⎨
⎧
⋅⋅
ω
ωω
ωω
ωωω
ω
ωωω
3
1
23
22
12111
3
2
1
F
HH
HH
HHH
X
XXX
nnn
n
n
X1
⋅⋅
X2
X3X4
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
Γ1
1
201ω12σ
Γ2
q21
202ω22σ
q31
203ω32σ
Γ3
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)
Frequency Response Functions are properties of linear dynamic systems
They are independent of the Excitation Function
Excitation can be a Periodic, Random or Transient function of time
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
)()()( fF
fXfH =
)()()(1 fG
fGfHFF
FX=
)()()(2 fG
fGfHXF
XX=
213 )( HHGG
GGfH
FX
FX
FF
XX ⋅=⋅=
2
1
22 *)( H
HGG
GG
GGG
fXX
FX
FF
FX
XXFF
FX =⋅=⋅=γ
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 H3
Input noise - Best -
Output noise Best - -
Input + output noise - - Best
Peaks (leakage) - Best -
Valleys (leakage) Best - -
Accuracy
)()()(1 fG
fGfHFF
FX= )()()(2 fG
fGfHXF
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)
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
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
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/s²] Time(Response) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s²]
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/s²] 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/s²]
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/s²)/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/s²)/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/s²)/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/s²)/N/s]InverseFFT⇒
ExcitationResponse
ForceMotion
InputOutputH ===
Modal Analysis 21
Hammer Test on Free-free Beam
Frequency
DistanceAmplitude FirstMode
Beam
SecondMode Third
Mode
ForceForceForceForceForceForceForceForceForceForceForce
Roving hammer method:Response measured at one point Excitation of the structure at a number of pointsby hammer with force transducer
Frequency Domain View
Modes of structure identified
FRF’s between excitation points and measurement point calculated
Press anywhereto advance animation
ForceModal
Domain
View
Acceleration
Modal Analysis 22
Measurement of FRF Matrix (SISO)
One rowOne Roving ExcitationOne Fixed Response (reference)
SISO
=
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3
: : :Xn Hn1 Hn2 Hn3...Hnn Fn
Modal Analysis 23
Measurement of FRF Matrix (SIMO)
More rowsOne Roving ExcitationMultiple Fixed Responses (references)
SIMO
=
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3
: : :Xn Hn1 Hn2 Hn3...Hnn Fn
Modal Analysis 24
FRF’s between excitation point and measurement points calculated
Whitenoiseexcitation
Shaker Test on Free-free Beam
Frequency
DistanceAmplitude FirstMode Second
ModeThirdMode
BeamAcceleration
Press anywhereto advance animation
Modal
Domain
ViewFrequency Domain View ∝ Force
Shaker method:Excitation of the structure at one pointby shaker with force transducerResponse measured at a number of points
Modes of structure identified
Modal Analysis 25
Measurement of FRF Matrix (Shaker SIMO)
One columnSingle Fixed Excitation (reference)
Single Roving Response SISOor
Multiple (Roving) Responses SIMOMultiple-Output: Optimize data consistency
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 H31 H32 H33...H 3n F3
: : :Xn Hn1 Hn2 Hn3...Hnn Fn
=
Modal Analysis 26
Why Multiple-Input and Multiple-Output ?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
Multiple-Output: Measure outputs at the same time in order to optimize data consistency
i.e. MIMO
Modal Analysis 27
Situations needing MIMO
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 DOF’s– One shaker moved to different reference DOF’s– MIMO
Modal Analysis 28
Measurement of FRF Matrix (MIMO)
=
More columnsMultiple Fixed Excitations (references)
Single Roving Response MISOor
Multiple (Roving) Responses MIMO
X1 H11 H12 H13 ...H1n F1
X2 H21 H22 H23 ...H2n F2
X3 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/s²)/N]Working : Input : Input : FFT Analyzer
0 200 400 600 800 1k 1,2k 1,4k 1,6k
100m
10
[Hz]
[(m/s²)/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/s²] 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/s²]
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/s²] Time(Response) - InputWorking : Input : Input : FFT Analyzer
0 40m 80m 120m 160m 200m 240m
-80
-40
0
40
80
[s]
[m/s²]
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/s²)/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/s²)/N/s]
InverseFFT
⇒Input
ExcitationResponse
ForceVibration
InputOutput
H(ω) = = =
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…
F1
a
F2
Modal Analysis 32
Impact Excitation
H11(ω) H12(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ H15(ω)⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅⋅ ⋅ ⋅ ⋅ ⋅
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)
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 durationa(t)
a(t)
ft
12
12
Modal Analysis 34
Weighting Functions for Impact Excitation
How to select shift and length for transient and exponential windows:
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)
Wτ
Correction of decay constant σ and damping ratio ζ:
Wm σ−σ=σ
WW
1τ
=σ
Wm0
W
0
m
0
ζ−ζ=ωσ
−ωσ
=ωσ
=ζ
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
Advantages:– Speed– No fixturing– No variable mass loading– Portable and highly suitable for
field work– relatively inexpensive
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
Conclusion– Best suited for field work
– Useful for determining shaker and support locations
Modal Analysis 38
Shaker Excitation
H11(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅H21(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
⋅⋅⋅
H51(ω) ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅ ⋅
Measuring one column of the FRF matrix by moving response transducer
# 1# 2
# 3# 4
# 5Accelerometer
ForceTransducer
VibrationExciter Power
Amplifier
LAN
Modal Analysis 39
Attachment of Transducers and Shaker
Accelerometer mounting:StudCementWax(Magnet)
Force Transducer and Shaker:StudStinger (Connection Rod)
Properties of StingerAxial Stiffness: HighBending Stiffness: Low
Accelerometer
ShakerFa
ForceTransducer
Advantages of Stinger: No Moment ExcitationNo Rotational Inertia LoadingProtection of ShakerProtection of TransducerHelps positioning of Shaker
Modal Analysis 40
Connection of Exciter and Structure
ExciterSlenderstinger
Measured structure
Accelerometer
Force Transducer
Tip mass, mShaker/Hammer
mass, M
Fs
Fm
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 ==
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
Modal Analysis 43
Swept Sine Excitation
AdvantagesLow Crest FactorHigh Signal/Noise ratioInput force well controlledStudy of non-linearitiespossible
DisadvantagesVery slowNo 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 ExcitationRandom signal:
– Characterized by power spectral density (GAA) and amplitude probability density (p(a))
a(t)
p(a)
Can be band limited according to frequency range of interestGAA(f) GAA(f)Baseband Zoom
Time
Signal not periodic in analysis time ⇒ Leakage in spectral estimates
Frequencyrange
Frequencyrange
Freq. Freq.
Modal Analysis 46
Random Excitation
AdvantagesBest linear approximation of systemZoomFair Crest FactorFair Signal/Noise ratio
DisadvantagesLeakageAveraging needed (slower)
Modal Analysis 47
Burst RandomCharacteristics of Burst Random signal :
– Gives best linear approximation of nonlinear system– Works with zoom
a(t)
Time
AdvantagesBest linear approximation of systemNo leakage (if rectangular time weighting can be used)
Relatively fast
DisadvantagesSignal/noise and crest factor not optimumSpecial time weighting might be required
Modal Analysis 48
Pseudo Random Excitation
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
Modal Analysis 49
Pseudo Random Excitation
T T T T
a(t)
Timep(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.
Modal Analysis 50
Pseudo Random Excitation
AdvantagesNo leakageFastZoomFair crest factorFair Signal/Noise ratio
DisadvantagesNo linear approximation of non-linear system
Modal Analysis 51
Multisine (Chirp)For sine sweep repeated every time record, Tr
Tr
A 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: The test time is longer than the test time using pseudo-random or random signal
Modal Analysis 53
Periodic Pulse
Leakage can be avoided using rectangular time weightingTransient and exponential time weighting can be used to increaseSignal/Noise ratioGating of reflections with transient time weightingEffects of non-linearities are not averaged outThe signal has a high crest factor
Rectangular, Hanning, or Gaussian pulse with user definable Δ t repeated with a user definable interval, Δ T
The line spectrum for a Rectangular pulse has a shaped envelope curve
sin xx
Δ t Δ T Time
1/Δ t Frequency
Special case of pseudo random signal
Modal Analysis 54
Periodic Pulse
Advantages
FastNo leakage(Only with rectangular weighting)
Gating of reflections(Transient time weighting)
Excitation spectrum follows frequency span in basebandEasy to implement
Disadvantages
No linear approximation of non-linear systemHigh Crest FactorHigh peak level might excite non-linearitiesNo ZoomSpecial 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 excitationFor study of non-linearities:
Random excitationFor slightly non-linear system:
Pseudo random excitationFor perfectly linear system:
Impact excitationFor field measurements:
Random impact excitationFor 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
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 every pair of resonances
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
Ιm ⏐Aij⏐
Re ⏐Aij⏐
ω
ω
Log Mag⏐Aij⏐
=
ω
Without rigidbody modes
Measured
Modal Analysis 62
Tests for Validity of Data: Coherence
Coherence)f(G)f(G
)f(G)f(
XXFF
2FX2 =γ
– 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:Noise in measured output signalNoise in measured input signalOther inputs not correlated with measured input signal
Bad measurements:LeakageTime varying systemsNon-linearities of systemDOF-jitterPropagation time not compensated for
Modal Analysis 64
Tests for Validity of Data: Linearity
Linearity X1 = H·F1
X2 = H·F2
X1+X2 = H·(F1 + F2)
a·X1 = H·(a· F1)⇒
– More force going in to the system will equate to
F(ω) X(ω)
more response coming out– Since FRF is a ratio the magnitude should be
the same regardless of input force
H(ω)
Modal Analysis 65
Tips and Tricks for Best ResultsVerify 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 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 FRFs…sometimes 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 FRFs…sometimes 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 FRFs…sometimes 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?Curve Fitting is the process of estimating the Modal Parameters from the measurements
|H|
ω
2σ
R/σ
ωd
0
-180Phase
Frequency
Find the resonant frequency
– Frequency where small excitation causes a large response
Find the damping – What is the Q of the
peak?
Find the residue– Essentially the ‘area
under the curve’
Modal Analysis 74
*
*
r
ijr
r r
ijr
rijrij pj
Rpj
RH)(H
−ω+
−ω==ω ∑∑
Residues are Directly Related to Mode Shapes!
Residues express the strength of a mode for each measured FRF
Therefore they are related to mode shape at each measured point!
Frequency
DistanceAmplitude FirstMode
Beam
SecondMode
ForceForceForceForceForceForceForceForceForceForceForceForce
Acceleration
ThirdMode
Modal Analysis 75
SDOF vs. MDOF Curve Fitters
ω
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)
Modal Analysis 76
Local vs. Global Curve Fitting
Local means that resonances, damping, and residues are calculated for each FRF first…then combined for curve fitting
Global means that resonances, damping, and residues are calculated across all FRFs
|H|
ω
Modal Analysis 77
Modal Analysis – Analyse Results1. Visually Inspect Data
– Look for obvious modes in FRF– Inspect ALL FRFs…sometimes 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 (ω)
Real Imaginary
Nyquist Log Magnitude
Modal Analysis 79
Which Curve Fitter Should Be Used?
Real Imaginary
Nyquist Log Magnitude
Frequency Response FunctionHij (ω)
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
ConclusionAll Physical Structures can be characterized by the simple SDOF model
Frequency Response Functions are the best way to measure resonances
There are three modal techniques available today: Hammer Testing, Shaker Testing, and Operational Modal
Planning and proper setup before you test can save time and effort…and ensure accuracy while minimizing erroneous results
There are many curve fitting techniques available, try to use the one that best fits your application
Modal Analysis 82
Literature for Further ReadingStructural Testing Part 1: Mechanical Mobility MeasurementsBrüel & Kjær Primer
Structural Testing Part 2: Modal Analysis and SimulationBrüel & Kjær Primer
Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J. EwinResearch Studies Press Ltd.
Dual Channel FFT Analysis (Part 1)Brüel & Kjær Technical Review # 1 – 1984
Dual Channel FFT Analysis (Part 1)Brüel & Kjær Technical Review # 2 – 1984
Modal Analysis 83
Appendix: Damping Parameters
3 dB bandwidthπσ
=Δ22f dB3
Loss factor0
dB3
0
db3
ff
Q1
ωωΔ
=Δ
==η
Damping ratio0
Bd3
0
dB3
2f2f
2 ωωΔ
=Δ
=η
=ζ
Decay constant
Quality factordB3
0
dB3
0
ffQ
ωΔω
=Δ
=
2f dB3
dB30ωΔ
=Δπ=ωζ=σ
σ=ωΔ 2dB3,
Modal Analysis 84
Appendix: Damping Parameters
( )tsineR2)t(h dt ω⋅⋅⋅= σ− , where the Decay constant is given by e-σt
The Envelope is given by magnitude of analytic h(t):
The time constant, τ, is determined 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
τ=σ
1
σ=τ
1
τπ=
ωσ
=ζ00 f2
1
τπ=ζ⋅=η
0f12
τπ=η
= 0f1Q
h(t)
Time
te σ−