Transcript Modal Analysis

Experimental
Modal Analysis
Modal Analysis 1
Shanghai Jiaotong University 2011
f(t)
x(t)
m
c
SDOF and MDOF Models
Different Modal Analysis
Techniques
k
Exciting a Structure
Measuring Data Correctly
=
+
+
++
Modal Analysis Post
Processing
Modal Analysis 2
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Simplest Form of Vibrating System
Displacement
d = D sinnt
Displacement
D
Time
T
Frequency
1
T
m
Period, Tn in [sec]
k
Frequency, fn=
n= 2  fn =
Modal Analysis 3
1
Tn
in [Hz = 1/sec]
k
m
Shanghai Jiaotong University 2011
Mass and Spring
time
n  2fn 
k
m  m1
m1
m
Increasing mass
reduces frequency
Modal Analysis 4
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Mass, Spring and Damper
time
Increasing damping
reduces the amplitude
m
k
Modal Analysis 5
c1 + c2
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Basic SDOF Model
f(t)
x(t)
m
c
k
Mx(t )  Cx (t )  Kx(t )  f (t )
M = mass (force/acc.)
C = damping (force/vel.)
K = stiffness (force/disp.)
Modal Analysis 6
x(t ) 
x (t ) 
x( t ) 
f (t ) 
Acceleration Vector
Velocity Vector
Displacement Vector
Applied force Vector
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SDOF Models — Time and Frequency Domain
F()
H()
X()
|H()|
1
k
f(t)

k
f (t )  mx(t )  cx (t )  kx(t )
Modal Analysis 7
1
c
x(t)
m
c
1
2m
 H()
0º
– 90º
– 180º
H ( ) 
0 =  k/m

X ( )
1

F ( )   2m  jc  k
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Modal Matrix
Modal Model
(Freq. Domain)
 X 1  
 X    H11   H 21   
 2   
H 22  
 X 3   
H 23  

 


  
   



  H n1  


X

 n 


X2
X3


  
H1n   
 




  F3  
 




 
  

H nn   




X4
X1
H22
H21
Modal Analysis 8
F3
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MDOF Model
Magnitude
1+2
d1 + d 2
m
2
1
Frequency
d1
Phase
dF
Frequency
0°
1
-90°
2
1+2
-180°
Modal Analysis 9
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Why Bother with Modal Models?
Physical Coordinates = CHAOS
Modal Space =
Simplicity
1
Rotor
1
21
Bearing
q1
2
01
2
Bearing
1
Foundation
22
q2
2
02
3
1
23
Modal Analysis 10
q3
2
03
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Definition of Frequency Response Function
F(f)
H(f)
X(f)
H
F
f
X
f
H
f
H(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)
Modal Analysis 11
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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 12
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Different Forms of an FRF
Compliance
(displacement / force)
Mobility
(velocity / force)
Inertance or Receptance
(acceleration / force)
Modal Analysis 13
Dynamic stiffness
(force / displacement)
Impedance
(force / velocity)
Dynamic mass
(force /acceleration)
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Alternative Estimators
H(f)
F(f)
X(f)
H( f )  X ( f )
F( f )
H1( f )  GFX ( f )
GFF ( f )
H 2 ( f )  GXX ( f )
GXF ( f )
H3 ( f ) 
GXX GFX

 H1  H 2
GFF GFX
2
GFX
G G*
H
2
 ( f )
 FX  FX  1
GFF GXX GFF GXX H2
Modal Analysis 14
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Alternative Estimators
F(f)
H(f)
SISO：
G (f)
H1( f )  FX
GFF ( f )
MIMO：


X 


X   F 




n s




 H 
H
n p
+
X(f)
N(f)


F 
p s
  N 
n s


H
  H F F H    N   F 


















G
H
G
G
n p ff
nf n p
nf n p
p p
1




  
G

 G 
H 
 ff 
 1 n p  xf 

 n p 
 n p
Modal Analysis 15




Shanghai Jiaotong University 2011
Alternative Estimators
+
F(f)
H(f)
X(f)
M(f)
SISO：
H 2 ( f )  GXX ( f )
GXF ( f )
MIMO：


X 


n s
X   X 
  H 
H
F  p s M n s 

n p






  H F X H  M X H 










G 
G

H 



G
 xx 
n n   n p  fx  p  n  mx  p  n 




Modal Analysis 16

H



2 n n








1
 G xx
G
n n fx n n




Shanghai Jiaotong University 2011
Which FRF Estimator Should You Use?
Accuracy
Definitions:
H1( f )  GFX ( f )
GFF ( f )
H 2 ( f )  GXX ( f )
GXF ( f )
Accuracy for systems with:
GXX GFX

GFF GFX
H3 ( f ) 
H1
H2
H3
-
Best
-
Best
-
-
Input + output noise
-
-
Best
Peaks (leakage)
-
Best
-
Valleys (leakage)
Best
-
-
Input noise
Output noise
User can choose H1, H2 or H3 after measurement
Modal Analysis 17
Shanghai Jiaotong University 2011
f(t)
x(t)
m
c
SDOF and MDOF Models
Different Modal Analysis
Techniques
k
Exciting a Structure
Measuring Data Correctly
=
+
+
++
Modal Analysis Post
Processing
Modal Analysis 18
Shanghai Jiaotong University 2011
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 19
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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 20
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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 21
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Frequency Response Function
[m /s²]
Tim e(R esp ons e) - Inpu t
Wo rkin g : Inpu t : In put : FFT Ana lyze r
80
40
0
Output Motion Response
H


Input
Force Excitation
FFT
-40
-80
0
40 m
80 m
12 0m
[s]
16 0m
20 0m
24 0m
Output
[m /s²]
Input
Au tosp ectrum(Res pon se) - Inp ut
Wo rkin g : Inpu t : In put : FFT Ana lyze r
10
1
10 0m
[(m /s²)/N]
Fre que ncy Res pon se H 1(R esp ons e,Excita tion) - In put (Mag nitu de)
Wo rkin g : Inpu t : In put : FFT Ana lyze r
10 m
1m
0
20 0
40 0
60 0
80 0
[H z]
1k
1,2 k
1,4 k 1,6 k
10 0m
[N ]
Au tosp ectrum(Exci tatio n) - Inpu t
Wo rkin g : Inpu t : In put : FFT Ana lyze r
1
Inverse
0
20 0
40 0
60 0
80 0
[H z]
1k
1,2 k
1,4 k 1,6 k
[(m /s²)/N/s
Im puls
] e R esp onse h1 (Res pon se,Excitatio n) - Inpu t (Re al Part)
Wo rkin g : Inpu t : In put : FFT Ana lyze r
2k

FFT
10
1k
0
-1k
-2k
0
40 m
80 m
12 0m
[s]
16 0m
20 0m
24 0m
10 0m
10 m
1m
10 0u
0
20 0
40 0
60 0
80 0
[H z]
1k
1,2 k
1,4 k 1,6 k
Frequency Domain
[N ]
Time Domain
FFT
Tim e(Excitation ) - Inpu t
Wo rkin g : Inpu t : In put : FFT Ana lyze r
20 0
10 0
0
-10 0
-20 0
0
40 m
80 m
Modal Analysis 22
12 0m
[s]
16 0m
20 0m
24 0m
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Hammer Test on Free-free Beam
Roving hammer method:
 Response measured at one point
 Excitation of the structure at a number of points
by hammer with force transducer

FRF’s between excitation points and measurement point calculated

Modes of structure identified
Amplitude
First
Mode
Second
Mode
Third
Mode
Beam
Acceleration
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Modal Analysis 23
Press anywhere
to advance animation
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Measurement of FRF Matrix (SISO)
One row


One Roving Excitation
One Fixed Response (reference)
SISO
X1
X2
X3
:
Xn
Modal Analysis 24
=
H11 H12 H13 ...H1n
H21 H22 H23 ...H2n
H31 H32 H33...H 3n
:
Hn1 Hn2 Hn3...Hnn
F1
F2
F3
:
Fn
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Measurement of FRF Matrix (SIMO)
More rows


One Roving Excitation
Multiple Fixed Responses (references)
SIMO
X1
X2
X3
:
Xn
Modal Analysis 25
=
H11 H12 H13 ...H1n
H21 H22 H23 ...H2n
H31 H32 H33...H 3n
:
Hn1 Hn2 Hn3...Hnn
F1
F2
F3
:
Fn
Shanghai Jiaotong University 2011
Shaker Test on Free-free Beam
Shaker method:
 Excitation of the structure at one point
by shaker with force transducer
 Response measured at a number of points
 FRF’s between excitation point and measurement points calculated

Modes of structure identified
Amplitude
First
Mode
Second
Mode
Third
Mode
Beam
Acceleration
White
noise
excitation
Modal Analysis 26
 Force
Press anywhere
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to advance animation
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
X2
X3
:
Xn
Modal Analysis 27
=
H11 H12 H13 ...H1n
H21 H22 H23 ...H2n
H31 H32 H33...H 3n
:
Hn1 Hn2 Hn3...Hnn
F1
F2
F3
:
Fn
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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 distributed
and
– avoid non-linear behaviour

Multiple-Output: Measure outputs at the same time in
order to optimize data consistency
i.e. MIMO
Modal Analysis 28
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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 29
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Measurement of FRF Matrix (MIMO)
More columns

Multiple Fixed Excitations (references)

Single Roving Response
MISO
or

Multiple (Roving) Responses
X1
X2
X3
:
Xn
Modal Analysis 30
=
MIMO
H11 H12 H13 ...H1n
H21 H22 H23 ...H2n
H31 H32 H33...H 3n
:
Hn1 Hn2 Hn3...Hnn
F1
F2
F3
:
Fn
Shanghai Jiaotong University 2011
Modal Analysis
(classic):
FRF
= Response/Excitation
Operational
Modal
Analysis
(OMA):
Response only!
[m/s²]
Time(Response) - Input
Working : Input : Input : FFT Analyzer
80
40
0
-40
-80
0
40m
80m
120m
[s]
160m
200m
240m
FFT
Output
Frequency Domain
[m/s²]
Time Domain
Autospectrum(Response) - Input
Working : Input : Input : FFT Analyzer
10
Inverse
FFT
1
100m
Frequency Response H1(Response,Excitation) - Input (Magnitude)
[(m/s²)/N]
10m
[(m/s²)/N/s]
Impulse Response h1(Response,Excitation) - Input (Real Part)
Working : Input : Input : FFT Analyzer
Working : Input : Input : FFT Analyzer

2k
1m
10
0
200
400
600
800
[Hz]
1k
1,2k
1,4k
1,6k
1k
0
100m
Input
[N]
-1k
Autospectrum(Excitation) - Input
Working : Input : Input : FFT Analyzer
1
-2k
0
100m
Natural Excitation
1m
0
FFT
Time(Excitation) - Input
Working : Input : Input : FFT Analyzer
200
100
0
-100
-200
0
40m
80m
120m
[s]
Modal Analysis 31
160m
200m
240m
400
600
800
[Hz]
1k
1,2k
1,4k
1,6k
0
40m
80m
120m
[s]
160m
200m
240m
10m
100u
[N]
200
200
400
600
800
[Hz]
1k
1,2k
1,4k
1,6k
Frequency
Response
Function
Impulse
Response
Function
Response
Output
Vibration
=
H() =
=
Input
Force
Excitation
Shanghai Jiaotong University 2011
f(t)
x(t)
m
c
SDOF and MDOF Models
Different Modal Analysis
Techniques
k
Exciting a Structure
Measuring Data Correctly
=
+
+
++
Modal Analysis Post
Processing
Modal Analysis 32
Shanghai Jiaotong University 2011
The Eternal Question in Modal…
F1
a
F2
Modal Analysis 33
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Impact Excitation
Measuring one row of the FRF matrix by
moving impact position
#5
#4
Accelerometer
#3
#2
H11() H12()       H15()





#1
LAN
Force
Transducer
Modal Analysis 34
Impact
Hammer
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Impact Excitation
a(t)
t

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)
GAA(f)
1
2
2
t
Modal Analysis 35
1
f
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Weighting Functions for Impact Excitation

How to select shift and length
for transient and exponential
windows:
Criteria
Transient weighting of the input signal
Exponential weighting
of the output signal

Leakage due to exponential time
weighting on response signal is well
defined and therefore correction of the
measured damping value is often possible
Modal Analysis 36
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Compensation for Exponential Weighting
b(t)
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
1
Window function
Original signal
Time
Weighted signal
shift
Length =  W
Record length, T
Correction of decay constant  and damping ratio z:
z
  m  W
Correct
value
Modal Analysis 37
Measured
value
W 
 m  W


 zm  z W
0 0 0
1
W
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Range of hammers
Description
Application
12 lb Sledge
Building and
bridges
3 lb Hand
Sledge
Large shafts
and larger
machine tools
Car framed
and machine
tools
1 lb hammer
Modal Analysis 38
General
Purpose, 0.3
lb
Components
Mini Hammer
Hard-drives,
circuit boards,
turbine blades
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Impact hammer excitation

Advantages:
–
–
–
–
Speed
No fixturing
No variable mass loading
Portable and highly suitable for
field work
– relatively inexpensive


Conclusion
– Best suited for field work
– Useful for determining shaker
and support locations
Modal Analysis 39
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
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Shaker Excitation
Measuring one column of the FRF matrix by
moving response transducer
Accelerometer
#5
#4
#3
#2
H11()               
H21()               



H51()               
#1
Force
Transducer
LAN
Vibration
Exciter
Modal Analysis 40
Power
Amplifier
Shanghai Jiaotong University 2011
Attachment of Transducers and Shaker
a
Accelerometer mounting:
 Stud
 Cement
 Wax
 (Magnet)
Force Transducer and Shaker:
 Stud
 Stinger (Connection Rod)
Modal Analysis 41
F
Force
Transducer
Shaker
Accelerometer
Properties of Stinger
Axial Stiffness: High
Bending Stiffness: Low
Advantages of Stinger:
 No Moment Excitation
 No Rotational Inertia Loading
 Protection of Shaker
 Protection of Transducer
 Helps positioning of Shaker
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Connection of Exciter and Structure
Force Transducer
Measured structure
Slender
stinger
Exciter
Accelerometer
Force and acceleration measurements unaffected by stinger compliance, but ...
Minor mass correction required to determine actual excitation
Fs
Fm

Fs  (m  M) X
Structure

Fm  M X
Tip mass, m
Piezoelectric
material
Modal Analysis 42
Shaker/Hammer
mass, M
mM
Fs  Fm
M
Shanghai Jiaotong University 2011
Shaker Reaction Force
Reaction by
external support
Reaction by
exciter inertia
Example of an
improper
arrangement
Structure
Suspension
Structure
Suspension
Exciter
Support
Modal Analysis 43
Exciter
Suspension
Shanghai Jiaotong University 2011
Sine Excitation
a(t)
A
RMS
Time
A
Crest factor  RMS  2
B(f1)

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
Modal Analysis 44
A(f1)
Shanghai Jiaotong University 2011
Swept Sine Excitation
Advantages




Low Crest Factor
High Signal/Noise ratio
Input force well controlled
Study of non-linearities
possible
Disadvantages


Modal Analysis 45
Very slow
No linear approximation of
non-linear system
Shanghai Jiaotong University 2011
Random Excitation
a(t)
Time
Random variation of amplitude and phase
 Averaging will give optimum linear
estimate in case of non-linearities
GAA(f), N = 1
GAA(f), N = 10
Freq.
Modal Analysis 46
System
Output
B(f1)
Freq.
A(f1)
System
Input
Shanghai Jiaotong University 2011
Random Excitation

Random signal:
– Characterized by power spectral density (GAA) and
amplitude probability density (p(a))
a(t)
p(a)

Time
Can be band limited according to frequency range of interest
GAA(f)
GAA(f)
Baseband
Zoom
Freq.
Frequency
range

Freq.
Frequency
range
Signal not periodic in analysis time  Leakage in spectral estimates
Modal Analysis 47
Shanghai Jiaotong University 2011
Random Excitation
Advantages




Best linear approximation of system
Zoom
Fair Crest Factor
Fair Signal/Noise ratio
Disadvantages


Modal Analysis 48
Leakage
Averaging needed (slower)
Shanghai Jiaotong University 2011
Burst Random

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
Modal Analysis 49
Shanghai Jiaotong University 2011
Pseudo Random Excitation

Pseudo random signal:
– Block of a random signal repeated every T
a(t)
T

T
T
T
Time
Time period equal to record length T
– Line spectrum coinciding with analyzer lines
– No averaging of non-linearities
GAA(f), N = 1
GAA(f), N = 10
Freq.
Modal Analysis 50
System
Output
B(f1)
Freq.
A(f1)
System Input
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Pseudo Random Excitation

Pseudo random signal:
– Characterized by power/RMS (GAA) and amplitude probability density (p(a))
a(t)
p(a)
T

T
T
T
Time
Can be band limited according to frequency range of interest
GAA(f)
GAA(f)
Baseband
Zoom
Freq.
Freq. range
Freq.
Freq. range
Time period equal to T
 No leakage if Rectangular weighting is used
Modal Analysis 51
Shanghai Jiaotong University 2011
Pseudo Random Excitation
Advantages





No leakage
Fast
Zoom
Fair crest factor
Fair Signal/Noise ratio
Disadvantages

Modal Analysis 52
No linear approximation of
non-linear system
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Multisine (Chirp)
For sine sweep repeated every time record, Tr
Tr
Time
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 smooth
Applications:
– Measurement of structures with non-linear behaviour
Modal Analysis 53
Shanghai Jiaotong University 2011
Periodic Random
A combined random and pseudo-random signal giving an
excitation signal featuring:
– No leakage in analysis
– Best linear approximation of system
A
A
A
T
T
T
B
B
B
C
C
C
Pseudo-random signal
changing with time:
transient response
steady-state response
Analysed time data:
(steady-state response)
A
B
C
Disadvantage:
 The test time is longer than the test time
using pseudo-random or random signal
Modal Analysis 54
Shanghai Jiaotong University 2011
Periodic Pulse
Special case of pseudo random signal
Rectangular, Hanning, or Gaussian pulse with user definable
D t repeated with a user definable interval, D T
Dt
DT
Time
The line spectrum for a
sin x
Rectangular pulse has a x
shaped envelope curve
1/D t





Frequency
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
Modal Analysis 55
Shanghai Jiaotong University 2011
Periodic Pulse
Advantages



Disadvantages
Fast
No leakage

(Only with rectangular weighting)

Gating of reflections

(Transient time weighting)


Excitation spectrum follows
frequency span in baseband
Easy to implement
Modal Analysis 56


No linear approximation of
non-linear system
High Crest Factor
High peak level might excite
non-linearities
No Zoom
Special time weighting might
be required to increase
Signal/Noise Ratio . This can
also introduce leakage
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Guidelines for Choice of Excitation Technique

For study of non-linearities:
Swept sine excitation

For slightly non-linear system:
Random excitation

For perfectly linear system:
Pseudo random excitation

For field measurements:
Impact excitation

For high resolution field measurements:
Random impact excitation
Modal Analysis 57
Shanghai Jiaotong University 2011
f(t)
x(t)
m
c
SDOF and MDOF Models
Different Modal Analysis
Techniques
k
Exciting a Structure
Measuring Data Correctly
=
+
+
++
Modal Analysis Post
Processing
Modal Analysis 58
Shanghai Jiaotong University 2011
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 Mobility
Central Decade
Transfer Mobility
Expanded
Frequency
Modal Analysis 59
Frequency
Shanghai Jiaotong University 2011
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
–
–
–
–
–
Modal Analysis 60
Accelerometer mounting is CRITICAL!
Uni-axial vs. Triaxial
Make sure DOF orientation is correct
Mount device under test...mounting will affect measurement!
Calibrate system
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Where Should Excitation Be Applied?
Hij 
Driving Point
Measurement
1
i
j
Transfer
Measurement
Modal Analysis 61
X  HF
2
i=j
ij
j
Xi Re sponse " i "

Fj Excitation " j "
i
 X 1  H11 H12  F1 
 
  F 
X
H
H
22   2 
 2   21
X1 H11  F1  H12  F2
X 2 H21  F1  H22  F2
Shanghai Jiaotong University 2011
Check of Driving Point Measurement
Im Hij

All peaks in
 (f ) 
 ( f ) 
X
X
 X( f ) 
Im 
, Re 
 and Im 


F
(
f
)
F
(
f
)
F
(
f
)






Mag Hij

An anti-resonance in Mag Hij
must be found between every
pair of resonances
Phase Hij

Phase fluctuations must be
within 180
Modal Analysis 62
Shanghai Jiaotong University 2011
Driving Point (DP) Measurement
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 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
Log MagAij
Measured

=
Re Aij

Modal Analysis 63
Without rigid
body modes

Shanghai Jiaotong University 2011
Tests for Validity of Data: Coherence
Coherence
 2( f ) 
GFX ( f )
2
GFF ( f ) G XX ( f )
– 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 64
Shanghai Jiaotong University 2011
Reasons for Low Coherence
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
Modal Analysis 65
Shanghai Jiaotong University 2011
Tests for Validity of Data: Linearity
Linearity
X1 = H·F1
X2 = H·F2

X1+X2 = H·(F1 + F2)
a·X1 = H·(a· F1)
H()
F()
X()
– More force going in to the system will equate to
more response coming out
– Since FRF is a ratio the magnitude should be
the same regardless of input force
Modal Analysis 66
Shanghai Jiaotong University 2011
Tips and Tricks for Best Results

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 of
experimental modal analysis!
Modal Analysis 67
Shanghai Jiaotong University 2011
f(t)
x(t)
m
c
SDOF and MDOF Models
Different Modal Analysis
Techniques
k
Exciting a Structure
Measuring Data Correctly
=
+
+
++
Modal Analysis Post
Processing
Modal Analysis 68
Shanghai Jiaotong University 2011
From Testing to Analysis
H( f )
Measured
FRF
Frequency
Curve Fitting
(Pattern Recognition)
H( f )
Modal Analysis
Frequency
Modal Analysis 69
Shanghai Jiaotong University 2011
From Testing to Analysis
H( f )
Modal Analysis
Frequency
m
SDOF Models
c
Modal Analysis 70
f(t)
x(t)
k
f(t)
x(t)
m
c
k
f(t)
x(t)
m
c
k
Shanghai Jiaotong University 2011
Mode Characterizations
All Modes Can Be Characterized By:
1. Resonant Frequency
2. Modal Damping
3. Mode Shape
Modal Analysis 71
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Modal Analysis – Step by Step Process
1. 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
Shanghai Jiaotong University 2011
Modal Analysis – Inspect Data
1. 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
Shanghai Jiaotong University 2011
Modal Analysis – Curve Fitting
1. 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 74
Shanghai Jiaotong University 2011
How Does Curve Fitting Work?

Curve Fitting is the process of estimating the Modal
Parameters from the measurements
2
H

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’
R/
d

0
Phase
-180
Frequency
Modal Analysis 75
Shanghai Jiaotong University 2011
Residues are Directly Related to Mode Shapes!
Hij ()   Hijr  
r
Amplitude
r
Rijr
j  pr
First
Mode
*

Rijr

Residues express the
strength of a mode for each
measured FRF

Therefore they are related
to mode shape at each
measured point!
j  pr*
Second
Mode
Third
Mode
Beam
Acceleration
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Force
Modal Analysis 76
Shanghai Jiaotong University 2011
SDOF vs. MDOF Curve Fitters

Use SDOF methods on LIGHTLY COUPLED modes

Use MDOF methods on HEAVILY COUPLED modes

You can combine SDOF and MDOF techniques!
SDOF
(light coupling)
MDOF
(heavy coupling)
H

Modal Analysis 77
Shanghai Jiaotong University 2011
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

Modal Analysis 78
Shanghai Jiaotong University 2011
Modal Analysis – Analyse Results
1. 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 79
Shanghai Jiaotong University 2011
Which Curve Fitter Should Be Used?
Frequency Response Function
Hij ()
Real
Nyquist
Modal Analysis 80
Imaginary
Log Magnitude
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Which Curve Fitter Should Be Used?
Frequency Response Function
Hij ()
Real
Nyquist
Modal Analysis 81
Imaginary
Log Magnitude
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Modal Analysis and Beyond
Experimental
Modal Analysis
Dynamic Model
based on
Modal Parameters
F
Structural
Modification
Hardware Modification
Resonance Specification
Modal Analysis 82
Response
Simulation
Simulate Real
World Response
Shanghai Jiaotong University 2011
Conclusion

All 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 83
Shanghai Jiaotong University 2011
Literature for Further Reading

Structural Testing Part 1: Mechanical Mobility Measurements
Brüel & Kjær Primer

Structural Testing Part 2: Modal Analysis and Simulation
Brüel & Kjær Primer

Modal Testing: Theory, Practice, and Application, 2nd Edition by D.J.
Ewin
Research 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 84
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Appendix: Damping Parameters
Modal Analysis 85
2
, D3dB  2
2
3 dB bandwidth
Df3 dB 
Loss factor

1 Df3 db D3 dB


Q
f0
0
Damping ratio
z
 Df3 dB D3 dB


2
2f0
20
Decay constant
  z 0   Df3 dB
Quality factor
Q
D3 dB

2
f0
0

Df3 dB D3 dB
Shanghai Jiaotong University 2011
Appendix: Damping Parameters
h(t )  2  R  et  sindt  , where the Decay constant is given by e-t

~
The Envelope is given by magnitude of analytic h(t): h( t )  h2 ( t )  h 2 ( t )
1

1



Decay constant
Time constant :
Damping ratio
Loss factor
Quality
Modal Analysis 86

1

0 2f0 
1
  2z 
f0 
z
Q
1
 f0 

h(t)
e  t
Time
The time constant, , 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
Shanghai Jiaotong University 2011