Modal Analysis Basic Theory

Modal Analysis
Gub – LMS Engineering Services
Modal Analysis
Basic Theory

2 copyright LMS International - 2005
 Equation of motion:
 Applied to SDOF:
 Applied force?
 Pulse (transient)
 Harmonic function (steady state)
 Combination
SDOF System Theory
ground
m
c
k
x(t)
f(t)
spring force + damping force + applied force = mass X acceleration
F = m X a

ωn
k
m

2
2
1,2
*
1 1 1 1 1 1
*
1 1
1
*
1 1 1
( ) 1/
( )
( ) ( / ) ( / )
( /(2 )) ( /(2 ) ( / )
1/
( )
( ) ( ) 2
X p M
FRF H p
F p p C M p K M
C M C M K M
j j
A A M
H p A
p p j

     
  
  
 
   
   
  
 
System poles
Residue
Damping factor - damped
natural frequency
Un-damped natural frequency ( C=0)
Laplace domain
SDOF System Theory
SDOF System Theory

SDOF
Frequency response
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10
-1
10
0
10
1
β
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
20
40
60
80
100
120
140
160
180
degrees
90 degrees
β
Magnitude Phase
f f
x
F

SDOF
Frequency response
Magnitude Phase
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
10
-1
10
0
10
1
10
2
zeta (z)
1%
2%
5%
10%
20%
50%
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
20
40
60
80
100
120
140
160
180
degrees
Zeta (z)
1%
2%
5%
10%
20%
50%
x
F

SDOF
Frequency response
 Magnitude & Phase  Real & Imaginary
0 2 4 6 8 10 12 14 16 18 20
10
-2
10
-1
10
0 Frequency Response Function
Frequency Hz
Log-Magnitude
0 2 4 6 8 10 12 14 16 18 20
-200
-150
-100
-50
0
Frequency Hz
Phase
0 2 4 6 8 10 12 14 16 18 20
-0.2
-0.1
0
0.1
0.2
Frequency Response Function
Frequency Hz
Real
Part
0 2 4 6 8 10 12 14 16 18 20
-0.4
-0.3
-0.2
-0.1
0
Frequency Hz
Imaginary
Part
x
F
x
F

Influence M, C, K
Stiffness K   resonance freq 1 
x
F

Influence M, C, K
Stiffness M   resonance freq 1 
x
F

Influence M, C, K
Stiffness C   amplitude  (resonance freq 1 )
x
F

SDOF
Frequency response
 Real vs. Imaginary
-0.2 -0.15 -0.1 -0.05 0 0.05 0.1 0.15 0.2
-0.35
-0.3
-0.25
-0.2
-0.15
-0.1
-0.05
0
Frequency Response Function
Real
Imaginary

SDOF
Frequency response
 FRF and system
parameters
0 2 4 6 8 10 12 14 16 18 20
10
-2
10
-1
10
0 Frequency Response Function
Frequency Hz
Log-Magnitude
0 2 4 6 8 10 12 14 16 18 20
-200
-150
-100
-50
0
Frequency Hz
Phase
damping controlled region
stiffness controlled region
mass controlled region
x
F
x
F

SDOF
Impulse Response
 FRF
 Inverse Fourier transform
= Impulse Response
 Poles
( )
( ) ( )
A A
H j
j j


  
 
( ) t t
h t Ae A e

  
 
* 2
, 1
n n d
j j
   z   z     
time [s]
amplitude
d
ω
/
π
2
time [s]
amplitude
d
ω
/
π
2
Amplitude
Time (s)

MDOF System Theory
ground
m 1
c
1
k
1
f
1
(t)
m 2
m n
ground
k
n+1
k
2
c
2
c
n+1
f
2
(t) f
n
(t)
x
1
(t) x
2
(t) x
n
(t)

MDOF System Theory
Equations of Motion
 Newton
 Applied to free body
 Assembled
 Shorthand notation
Free Body Diagram
m n
fn(t)
x
n
(t)
k
n+1
c
n+1
k
n
c
n
n
n
x
Forces mx


1 1 1 1 1 1
( ) ( ) ( ) ( ) ( )
n n n n n n n n n n n n n n n
k x x k x x c x x c x x f t m x
     
        
1 1 1 2 2 1 1 2 2 1
2 2 2 2 3 3 2 2 2 3 3 2
1 1
0 0 0 0
0 0
0 0 0 0 0 0
n n n n n n n n
m x c c c x k k k x
m x c c c c x k k k k x
m x c c x k k x
 
   
           
           
     
           
 
          
          
 
          
           
1
2
n
f
f
f
 
 
 

  
  
  
 
( ) ( ) ( ) ( )
M x t C x t K x t f t
  

MDOF
Physical Space vs. Modal Space
 Physical space
 Modal space

( ) ( ) ( ) ( )
T
i i i
m p t c p t k p t f t
     
   
     
     
ground
m1
c1
k1
f1(t)
m2 mn
ground
kn+1
k2
c2
cn+1
f2(t) fn(t)
x
1(t) x
2(t) x
n(t)
( ) ( ) ( ) ( )
  
Physical
Mass
Damping
Stiffness
Modal
Mass
Damping
Stiffness
mode 1
m1
c1
k1
p(t)
1
mode 2
m2
c2
k2
p(t)
2
mode n
mn
cn
kn
p(t)
n

2 1
( ) ( ) ( )
( ) [ ]
X p H p F p
H p p M pC K 

  
2
( ) ( ) ( )
p M pC K X p F p
  
MDOF
Transfer function
 Time-domain equation of motion
 Laplace domain
 Transfer Function
 Poles & residues
 Modal scaling factor
( ) ( ) ( ) ( )
  
*
*
1
( )
n
k k
k k k
A A
H p
p p

 
   

{ } T
k k k k
A Q
    
k
Q
Non-trivial
mathematics

MDOF
FRF
 Magnitude & Phase  Real & Imaginary

MDOF
Modal Decomposition
mode 1
m
1
c
1
k
1
p(t)
1
mode 1
m
1
c
1
k
1
p(t)
1
mode 1
m
1
c
1
k
1
p(t)
1
mode 2
m
2
c
2
k
2
p(t)
2
mode 2
m
2
c
2
k
2
p(t)
2
mode 2
m
2
c
2
k
2
p(t)
2
mode 3
m
3
c3
k
3
p(t)
3
mode 3
m
3
c3
k
3
p(t)
3
mode 3
m
3
c3
k
3
p(t)
3
*
,1 ,1
*
1 1
*
,2 ,2
*
2 2
*
,3 ,3
*
3 3
( )
pq pq
pq
pq pq
pq pq
A A
H j
j j
A A
j j
A A
j j
  
 
 
 
 
 

MDOF
Impulse Responses
 FRF
 Inverse Fourier transform
= Impulse Responses *
*
, ,
1
( ) k k
n
t t
pq pq k pq k
k
h t A e A e
 

 

*
, ,
*
1
( )
n
pq k pq k
pq
k k k
A A
H j
j j

  
   


Deformation at certain moment = linear
combination of mode shapes
Linear combination factors depend on input
forces, frequency, damping and mode
shape at input locations
Vibration
Response
Mode shapes
=
+ + + + ...
a1
x x x x
a2 a3 a4
Real structures
Modal decomposition

Modal Analysis
Modal Analysis
Parameter Estimation

Overview
1. Selection of method
2. Selection of measurements
3. Selection of frequency band
4. Selection of time block
5. Estimation of number of poles
6. Estimation of poles
7. Estimation of modal vectors
Modal Analysis Process

0.00 80.00
Hz
10.0e-6
0.10
Log
(
g/N
)
0.00 80.00
Linear
Hz
0.00 80.00
Hz
-180.00
180.00
Phase
°
0.00 6.00
s
-1.07
0.91
Real
(
g/N
)
Modal Parameter estimation
Frequency domain versus Time domain
Inverse
Fourier
transform
Frequency
domain
Time
domain
FRF IRF
*
*
1
( )
n
k k
k k k
A A
H
j j

  
 

*
*
1
( ) e e
n
k k
k k
k
t t
h t A A

 
 

{ } T
k k k k
A Q
     * 2
, 1
k k k k k k
j
   z   z 

Modal Parameter Estimation
SDOF vs. MDOF
SDOF
Methods
MDOF
methods

one global estimate for f
!Consistency of data is important!
Local vs. Global vs. Polyreference Estimates
f = 136.07 Hz
f = 135.74 Hz
Local Global Polyreference
Separation of
repeated poles!

Single Degree of Freedom Techniques
 FRF around (lightly-damped and not multiple) pole:
!Assumes at resonance only one mode is important!
 Method (Process)
 Find resonance frequencies
 Estimate residues
 Estimate damping ratios
*
*
1
( )
n
k k
k k k
A A
H
j j

  
 
 ( ) k k
k
k k k k
A A
H
j
  
   z 

Single Degree of Freedom Techniques
• Estimation of resonance
frequencies
• Peak Picking
• Circle fit
• Very simple to use, BUT
limited applicability
• Well-separated
modes
• Very low damping
• Interesting when strong
mass loading effects

SDOF – Estimation of Damping
 Half-power bandwidth method

3 dB
1 2
z
 



2 1
2

3D FRF & Circle Fit
-10
0
10
20
30
40
50
-0.1
0
0.1
0.2
0.3
-0.1
-0.05
0
0.05
0.1
0.15
0.2
0.25
Imag
Real
Freq

Modal Parameter Estimation - MDOF
Complex Mode Indicator Function (CMIF)
 The rank of the FRF matrix at a certain frequency depends on the number of modes having a
significant contribution at that frequency
E.g. around (lightly-damped and not multiple) pole
 Numerical tool to asses the rank of a matrix
Singular Value Decomposition (SVD)
* *
*
1
{ } { }
( )
T H
n
k k k k k k
k k k
Q Q
H
j j

       
  
 

( ) { } T
k k k
H     
Rank = 1, since column vector * row vector
( ) ( ) ( ) ( )
H
H U V
     

( ) ( ) ( ) ( )
H
H U V
     
 SVD of FRF @ all spectral lines
CMIF =
Singular values as
a function of
frequency
(diagonals of )
( )
 
Double pole

 Cada-X: estimation of complete set of
modal parameters
 Test.Lab: “mode indicator”, background of
stabilization diagram

Overview
1. Selection of method
2. Selection of measurements
3. Selection of frequency band
4. Selection of time block
5. Estimation of number of poles
6. Estimation of poles
7. Estimation of modal vectors
Modal Analysis Process

How many modes to estimate?
 Major problem in modal parameter estimation
What is the model order?
How many modes to curve-fit?
 Solutions
Sum of frequency response functions
Mode indicator functions
Stabilisation diagram
(Error chart)
*
*
1
( )
n
k k
k k k
A A
H
j j

  
 

0.00 80.00
Hz
10.0e-6
0.10
Log
(
(m/s2)/N
)
22.56 41.19

Mode indicator functions
 “Sum” of FRFs
 Multivariant mode indicator function (MvMIF)
 Complex mode indicator function (CMIF)
22.00 42.00
Linear
Hz
4.73e-3
0.09
Log
(
)
22.00 42.00
Linear
Hz
22.00 42.00
Hz
-180.00
180.00
Phase
°
22.00 42.00
Hz
0.00
1.00
Real
/
22.00 42.00
Hz
1.01e-3
1.00
Log
/

Stability
: new
: freq
: damp + freq
: part. vector + freq
: all
o
f
d
v
s
Stabilisation diagram
 Try a whole range of model orders
 Compare modal parameters at current order with
previous order
*
*
1
( )
n
k k
k k k
A A
H
j j

  
 

n

Stabilisation diagram
Model order problem shifted to problem of separating
true from computational poles?
!Difficulty: Avoid Computational Modes!

Frequency-Domain Curve-Fitting
 Pole-residue models not directly used
 Non-linear optimisation problem
 Right matrix-fraction model
 Can be linearised
Poles & participation factors
Mode shapes
    
     
"
)
(
)
(
)
(
)
(
)
(
)
(
"
)
(
)
(
)
(
0
0
1
1
0
0
1
1
1














































j
j
j
j
j
j
A
B
H
p
p
p
p
p
p
p
p


 
r

 
r

Quotes have been introduced because
the ratio notation is normally not used
for matrices in the denominator
*
*
  
 












n
i i
H
i
i
i
T
i
i
j
l
v
j
l
v
H
1
*
*
}
{
}
{
)
(
measured unknowns
measured unknowns

LMS PolyMAX
Linear Least Squares
 Right matrix-fraction model (z-domain)
 Linearisation
By minimising the error (in a linear least squares sense), the model
can be found from the data
 Algorithm optimisation
“Reduced Normal Equations”
 Problem size reduction (Final dimension
not related to the number of DOFs: very
large-size problems can be tackled)
 Memory and speed optimisation
    
     
"
"
)
(
)
(
)
(
0
0
1
1
0
0
1
1
1
z
z
z
z
z
z
A
B
H
p
p
p
p
p
p
p
p










































    
)
(
)
(
)
(
error 



 A
H
B
 
 
 
 
 
0
)
)
(
(
1
0



















p
H
M

  
)
(
,
)
( 
 B
A  
)
(
H
t
j
z


 e


LMS PolyMAX
Consequence of z-Domain Description
 Properties of z-domain model with real-valued coefficients
 Consequence: pole estimation close to borders of frequency band less accurate
-40 -20 0 20 40 60 80
10
-3
10
-2
10
-1
Magnitude
-40 -20 0 20 40 60 80
-200
0
200
f [Hz]
Phase
[deg]
Complex conjugated and mirrored
)
(
)
(
)
(
)
2
(
*










H
H
H
t
H

LMS PolyMAX
Implementation
• Step 1: reduced normal equations
• Establish reduced normal equations for maximum
model order
 
 
 
 
 
0
)
)
(
(
1
0



















p
H
M


model order
 
 
 
 
 
0
)
)
(
(
1
0



















p
H
M

LMS PolyMAX
Implementation
• Step 2: stabilisation diagram
• Compute poles
and participation
factors for the
model order range
• Select stable
modes

model order
 
 
 
 
 
0
)
)
(
(
1
0



















p
H
M

LMS PolyMAX
Implementation
• Step 2: stabilisation diagram
• Compute poles
and participation
factors for the
model order range
• Select stable
modes
• Step 3: mode shapes and residuals
• Least-Squares Frequency-Domain (LSFD) Method
      UR
LR
j
l
v
j
l
v
H
n
i i
H
i
i
i
T
i
i















 

2
1
*
*
)
(
Selected
from
stabilisation
diagram
unknowns
measured

Least Squares Frequency Domain (LSFD)
 Interpretation of stabilization diagram yields
poles (freq+damp) and participation factors
 Second step to estimate mode shapes and
residuals = LSFD
0.00 80.00
Linear
Hz
100e-9
1.00
Log
(
)
FRF BACK:125:-Y / RAIL:151:+Y
Synthesized FRF BACK:125:-Y/RAIL:151:+Y
Synthesized FRF BACK:125:-Y/RAIL:151:+Y
0.00 80.00
Linear
Hz
0.00 80.00
Hz
-180.00
180.00
Phase
°
Blue: without residuals
      UR
LR
j
l
v
j
l
v
H
n
k k
H
k
k
k
T
k
k















 

2
1
*
*
)
(
Selected
from
stabilisation
diagram
unknowns
measured

Example
Porsche 911 Targa Carrera 4
 Full vehicle
Heavily damped
 Excitation
4 shakers
 CMIF
Complex mode indicator function

Example
Porsche 911 Targa Carrera 4 - Stabilisation
LSCE (least squares complex exponential)

FDPI (frequency domain direct parameter)
Example

PolyMAX (Z-domain estimation)
Example

 Satellite
Very lightly damped
 Excitation
5 shakers
 CMIF
Example
Radar sat

 “Textbook” stabilisation
 Analysis in one single
range possible with
PolyMAX
 Classical methods: only
possible through sub-
ranges
Successful identification of many closely-spaced local modes
Example
Radar sat - Stabilisation

Modal Analysis
Modal Analysis
Validation

Validation
 Synthesis of FRFs
 Plot and animate mode shapes
 MAC matrix
 Mode participation
 Mode colinearity
 Mode complexity
 …
0.00 80.00
Linear
Hz
10.0e-6
1.00
Log
(
(m/s2)/N
)
FRF BACK:125:-Y / FRNT:15:+Z
Synthesized FRF BACK:125:-Y/FRNT:15:+Z
0.00 80.00
Linear
Hz
0.00 80.00
Hz
-180.00
180.00
Phase
°

Porsche 911 Targa Carrera 4
PolyMAX FRF Synthesis
3.50 30.00
Linear
Hz
100e-6
10.0e-3
Log
(
g/N
)
FRF moto:9:+Z/karo:1:+Z
3.50 30.00
Linear
Hz
3.50 30.00
Hz
-180.00
180.00
Phase
°
3.50 30.00
Linear
Hz
10.0e-6
10.0e-3
Log
(
g/N
)
3.50 30.00
Linear
Hz
3.50 30.00
Hz
-180.00
180.00
Phase
°
3.50 30.00
Linear
Hz
10.0e-6
10.0e-3
Log
(
g/N
)
FRF moto:9:+Z/moto:2:+Z
3.50 30.00
Linear
Hz
3.50 30.00
Hz
-180.00
180.00
Phase
°
3.50 30.00
Linear
Hz
10.0e-6
10.0e-3
Log
(
g/N
)
3.50 30.00
Linear
Hz
3.50 30.00
Hz
-180.00
180.00
Phase
°

MAC Matrix
Spatial Aliasing
 PZL-Sokol Helicopter
MAC

Validation
Mode Complexity and Mode Participation
Mode Frequency MPC (%) MP(%)
FRNT:15:+Z
MP(%)
RAIL:151:+Y
MP(%)
1 26.996 Hz 99.968 65.783 100.000 23.634
2 27.346 Hz 99.903 100.000 2.688 1.768
3 31.555 Hz 99.954 100.000 51.330 12.264
4 34.538 Hz 99.778 100.000 5.110 12.763
5 35.948 Hz 99.906 100.000 12.841 11.651
6 37.872 Hz 99.871 100.000 27.591 12.127
7 49.369 Hz 99.950 100.000 32.962 9.651
8 52.728 Hz 99.308 100.000 21.333 1.047
9 53.432 Hz 99.541 100.000 10.176 0.514
10 54.058 Hz 96.061 88.338 100.000 7.671
11 54.902 Hz 99.439 27.366 100.000 1.806
12 55.349 Hz 98.500 18.353 100.000 2.671
13 58.430 Hz 98.395 100.000 63.563 2.433
Vertical shaker
Horizontal shaker
Vertical
shaker
Horizontal
shaker

Validation
Mode Colinearity
 Plot mode shape in complex
plane

Modal Analysis Basic Theory

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