Modal Parameter Extraction
Methods
Modal Analysis and Testing
S. Ziaei-Rad
Type of Modal Analysis
By domain
Frequency domain (FRFs)
Time domain (IRFs or response history)
1.
2.
By Frequency range
SDOF method
MDOF method
1.
2.
In this course
1.
2.
Single-FRF methods
Multi-FRF methods
Preliminary Checks of FRF
Data
Visual checks
1.
2.
3.
4.
5.
Low-frequency asymptotes
High-frequency asymptotes
Incidence of anti-resonances
Overall shape of FRF skeleton
Nyquist plot inspection
Basic Skeleton Theory
IF M1 IS ASYMPTOTIC TO Y
IS M 2 ALSO ASYMPTOTIC TO Y ?
Mobility Skeleton
Skeleton Geometry
 S 

BD  DE  DC  log
A 

 m1  m 2 
 S2 
Ym 1S



log

log


2 BD  EC  DC  log 2 
Ym1  m2 S
 m1 
 A 
s2 m2 m1  m2

s2  k1 m11  m12  2R
k1
m1


Skeleton Geometry
Mass-dominated characteristics
Stiffness-dominated characteristics
Abnormal characteristics
Assessment of Multiple-FRF Data
[ A]Lnp  [{H11 ( )}L1 {H12 ( )}L1  {Hnp ( )}L1 ]
[ A]Lnp  [U ]LL []Lnp [V ]
T
npnp
[P]Lnp  [U ]LL []Lnp
Principle Response Function (PRF)
Mode Indicator Functions (MIFs)
-The technique is used to determine the number of modes
present in a given frequency range, to identify repeated
natural frequencies and to pre-process the FRF data prior
to modal analysis.
Consider a set of FRF data from multiple excitation
measurements or from multi-reference impact tests typically
consists of an n  p matrix where:
n number of measurement DOFs
p number of excitation or reference DOFs
Complex Mode Indicator
Function (CMIF)
[H ()]n p  [U ()]nn [()]n p [V ()]Hp p
[CMIF( )]p p  [( )] pn [( )]n p
T
The CMIF is the squares of the singular values and are usually
plotted as a function of frequency in logarithmic form.
Natural frequencies are indicated by large values of the first CMIF.
Double modes by large values of second CMIF.
Other MIFs
MMIF:
 [ H R ] T [ H R ]  [ H I ] T [ H I ] { F }  [ H R ] T [ H R ]{ F } *
 results from the eigenvalue solution equation (*) for each frequency
And these values are plotted as a function of frequency.
The MMIF takes a value between 0 and 1, with the resonance
frequencies now identified by minimum values of MMIF instead of
Maximum values for the CMIF.
RMIF:
[H ] [H ]{F}  {F}

I
R
In this version, natural frequencies are identified by zero crossing of
the RMIF values.
MIFs
Complex Mode Indicator
Function (CMIF)
Multivariate Mode Indicator
Function (MMIF)
Modal Analysis Method
H jk
H jk
r   A
H

Curve Fit Analysis:
1- SDOF Methods
2- MDOF Methods
Modal Analysis
GIVEN: MEASURED
H(w)
FRF DATA:

MODEL:
N
H ( )  
r =1
Ar
r2   2 + i r
DETERMINE: BEST ESTIMATES FOR THE MODAL PARAMETERS
1 1 A1
 2  2 A2






SDOF Curve-fit Method
H jk
Im
H jk
Im
H jk
Im
Re
Re
Re
 3 , 3 , A 3
 2 , 2 , A 2

1 , 1 , A1
SDOF Modal Analysis
N
 jk () =

s
    is s
2
s
s=1
 jk () =
r
2
 jk () 
2
2


s=1
 r
r
s
A jk
2
  r
A jk
    is s
2
s
    ir  r
2
r
(1)
2
N
A jk
    ir  r
2
r
A jk
2
2
 r R jk
2
(3)
(2)
Complete FRF
Peak Amplitude Method
1- First, individual resonance peak are detected on the FRF plot
and the frequency of one of the maximum responses taken as
natural frequency of that mode  r .
2- The local maximum value of the FRF is noted Hˆ and the
frequency bandwidth of the function for a response level
Hˆ / 2 is determined (  ). The two points are thus identified as
a ,b (Half-power point)
3- The damping of the mode can now be estimated from one of the
2
2
following formulae.   a  b   , 2  
r
2r2
r
r
r
4- The modal constant can be found from:
Ar
ˆ
| H | 2
 r r
or
Ar | Hˆ |  r2r
Peak Amplitude Method




2
 b  r a
a   b
r 
r
Peak Amplitude Method
CASE (a)
CASE (b)
Limitation of Peak Amplitude Method
-The estimates of both damping and modal constant depend
heavily on the accuracy of maximum FRF level, while it is
not possible to measure this quantity with great accuracy.
-Most of the errors in measurement are around the resonance
region particularly for the lightly damped structures.
-Only real part of the modal constant can be calculated.
- The single mode assumption is not completely correct. Even
with clearly separated modes, it is often found that the
neighboring modes do contribute a noticeable amount to the
total response.
-A more general method called circle-fit method will introduce
in next section.
Circle-Fit Method
Properties of Modal Circle
-Here, we consider a system with the structural damping.
- Thus, we shall use the receptance form of FRF.
-As we said earlier, it is this parameter that produce an exact
circle in a Nyquist plot.
-If the structure possesses the viscous damping, then the mobility
type FRF should be used.
-Although, this later need a different general approach, most of the
following analysis and comment apply equally to that case
-Some modal analysis packages, offer the choice between the two
types of damping and simply take the mobility or receptance data
for the circle-fitting according to the selection.
Properties of Modal Circle
 ( ) 
Im
1
 r2 1  ( /  r ) 2  ir

tan 
Re

The effect of modal constant
is to scale and rotate the circle

r
(1  (  ) 2 )
r




   
tan    (1  (  ) 2 ) / r
r
r
  2   r 2 (1  r tan  )


    r 2 r 1+ 2 (1  (  ) 2 ) 
r
r


 2

 
 
 
 
2

2  r 2   2
=
r2
 2
   
r
    r 2 r

= 0,    r
Properties of Modal Circle
Im
Re

 b

 a
a
b
r
Consider two points
 a  b  90
 r 2   b2

r  r 2
tan   b
b
a2   r2

r  r 2
 a 2  b 2  

1



r  
2

1
1

 r
  tan 2  a + tan 2  b 
tan  a

 a  b  
2

 
1
1
  r   tan 2  a + tan 2  b 
HALF  POWER POINTS; 1 ,  2
r  
 2 2  12  2  1
r 

2
r
2 r
r
D jk 
r
A jk
 r2r
Circle-Fit Analysis Procedure
1- Select point to be used
2- Fit circle, calculate quality of fit
3- Locate natural frequency, obtain damping estimate
4- Calculate multiple damping estimates and scatter
5- Determine modal constant modulus and argument
Circle-Fit Analysis Procedure (Step 1)
Select point to be used
•Can be automatic selection or by the operator
judgment
•The selected point should not be influenced by
neighboring modes
•The circle arc should be around 270 degree (if
the second rule is not violated)
•Not less than six points should be used
SELECT DATA POINTS
Im()
Re( )
Circle-Fit Analysis Procedure (Step 2)
Fit circle, calculate quality of fit
•Different routins can be used to fit the circle (e.g. least-square deviation)
• At the end of this process, the centre and radius of the circle are specified.
•An example of the process is shown in next slide.
Im()
Im()
Re( )
Re( )

1



3

15
14
12
2
13
r
Circle-Fit Analysis Procedure (Step 3)
Locate natural frequency, obtain
damping estimate
Im()
•The radial lines from the circle centre to the
point around the resonace are drawn
•The sweep rate the can be calculated, then
natural frequency and damping ratio
1.The frequency of maximum response
2.The frequency of maximum imaginary
receptance
3.The frequency of zero real receptance
Re( )
12
13
15
14
(iii)
(i)
(ii)
Circle-Fit Analysis Procedure (Step 3)
Estimation of Natural Frequency
Circle-Fit Analysis Procedure (Step 4)
(Damping Estimate)
Im()
Re( )
b
b
a
12
13
a
15
14
-Using different points (one below and
one after resonance), a set of damping
ratio will be calculated.
-Ideally they should all be identical
-If deviation is less than 4 to 5 percent,
then we did a good analysis
-If the scatter is 20 to 30 percent, there
is something unsatisfactory.
-If the variation of damping is random,
is probably due to random noise
-If the variation is systematic, it is due to
systematic errors (set-up, effect of near
modes, non-linearity)
r
  2a   2b  

1
 

r  

  2r   tan(a / 2) + tan( b / 2) 
Circle-Fit Analysis Procedure (Step 4)
(Damping Estimate)
a
d
b
c
e
a- linear data
b- random noise
c- error in the data
d- modal analysis error
e- non-linearity
Inverse or Line-fit Method
1
k -2 m
d
 ( ) =
=
i
2
2
2
2
k - m + i d ( k - m )  ( d )
( k -2 m ) 2  ( d ) 2
 -1 (  ) = ( k -  2m ) + i ( d )
Standard FRF plot format
Or
 i( c )
Inverse FRF plot format
SDOF Modal Analysis Using Inverse FRF Data
GENERAL SDOF ASSUMPTION:
()
  r
=
Ar
 Rr
2
2
2
r    ir r
RESIDUAL EFFECTS
OF OTHER MODES
DEFINE:
' () = ()  ()
  ONE OF THE VALUES OF  NEAR  r
AND
 2  2 
() =

() 
' ()
2
r
AN ‘INVERSE’ FRF PARAMETER


  2  ir2r r2  2  ir2r
 Re()  i Im()
Ar
SDOF Modal Analysis Using Inverse FRF Data
() 
2r  2  i2r r  2r  2  i2r r 
Ar
 RE()  i IM()
RE()  m R  2  c R
WHERE


,

IM()  mI  2  c I
m R  a r  2   2r  b r  2r r




mI   b r  2   2r  a r  2r r

A r
 a r  i br 
Analysis Step One
From measured FRF near  r , fix one point ( at    j ) and
Calculate    for all other points. Plot and fit:
RE()


 

IM()






Slopes of best-fit lines for
RE    vs  2 
IM    vs  2 
 mR  j
ml  j

2








 
2
Analysis Step One
Note
mR  n R 2  d R
ml  nl 2  d l
Where
So
   
b r  2r   a r  2r r 
nR  a r ;
d R   b r  2r r  a r  2r
nI   b r ;
dl 
 2r  d R / (pr  1) n R
r  (q  p) / (1  pq)
a r   2r (pr  1) /
b r   p. a r
 (1  p2)
dR

nl
p
nR
q
dl
dR
ANALYSIS STEP TWO
- Repeat step one for all values of  j
(Compute (mR)1 , (mR) 2 , ... , (ml)1 , (ml) 2 , ... )
-
Plot (mR) vs 2 , (ml) vs 2
nR 2  d R
- Fit best straight line
nl 2  d l
- Find n R; d R; nl; d l
 Hence
From jth Plot:step One
mR
 mR  j
 r; r; a r; b r



Ml
 

j

SLOPE = n R
INTERCEPT = d R
2

 
 


 
 
SLOPE = nl
INTERCEPT = d l


2


Line Fit Modal Analysis
Line fit modal analysis
Plot of real and imaginary
Line fit modal analysis
a- Plot of Real and Imaginary
b- Slope from a
Regenerated
FRFs
~
~ () =

jk
m2

r = m1
r A jk
~ 2   2  i~
~ 2



r
r r
Measured and regenerated without
Residual effect
Measured and regenerated with
residuals
Residuals
m2
 jk () =
r A jk
2

2
2




i


r
r
r
r = m1
m 1 1


r=1
r A jk
2
N

LOW-FREQUENCY MODES
 2r    ir  r 2
r A jk
2

HIGH-FREQUENCY MODES
2
2




i


r
r r
r = m 2 1
m2
 jk () 

r = m1
 2r
r A jk
2
   i r  r
2

1

2
M Rjk

1
K Rjk

 

RESIDUALS
Representation of Residuals as Linear
Functions
Residuals
m2
Ar
1
H()   2
 2 RL  RH
2
2

r m1 r    ir r
m1 1
Ar
RL   2

2
r 1 r   
m1 1
 1  Low Frequency
  2 Ar Residual

 
r 1 
(L.F. Residual)
 Ar 
Ar
RH   2
   
2
r  m 2 1 r   
 r 
orN
High Frequency Residual
(H.F. Residual)
2
L.F. Residuals
(Rigid Body Modes)
f
M, I g
xa ,  a
d
Mxa  f
Ia  fd
z  xa  da
 1 d2 
   f
 M Ia 
Z
RL
z  1  1 d 2 
   2   
f    M Ia 
H.F. Residuals
R jk 
j k:
j k:
N

r  n 1
r
 j r k

2
r
ALL TERMS +VE ADDITIVE
SOME TERMS +VE, SOME -VE
 TENDENCY TO CANCEL
Modal Analysis Methods
Modal Analysis in Frequency Domain
MDOF Curve-fit Method
H

1
2
1
2
A1
A2






N
N
AN
Curve - Fitting In General
(Nonlinear Least-Squares)
MEASURED FRF DATA:
 jk (1) = 1m


 jk (1) = 1
THEORETICAL MODEL
FOR FRF DATA:

M2
1
1

sA
1    2
  2  R  2  R
2
K
1 M
s  M 1  s  1  i  s s

M
 1  1  1


p
E1  1 ; E  We Ee
2
e 1
 
E
 0; q  1A , 2 A , .... , etc
q
Modal Analysis Using Rational Fractions
USE ALTERNATIVE FORMAT FOR FRF:
b0  b1 (i )     b2 N 1 (i ) 2( N 1)
H() 
a0  a1 (i )     a2 N 2 N
INSTEAD OF PARTIAL FRACTION
H()  
or
Ar
2r  2  2i r r

Ar
Sr  i
Rational Fraction Curve Fits
~m ~m
H k  H (k )
LET
AND

 b0  b1 (i )     
~m


ek  
 Hk

 a0  a1 (i )        k
ek  b0  b1 (ik )     b2m2 (ik )
2( m1)
E k  ek


~m
 Hk a0  a1 (ik )     (ik )2m
2
~m
GIVEN SEVERAL VALUES OF k , Hk
FIND
a 0, a1, ... b n TO MINIMISE
E
k

Rational Fraction Curve Fits
b0 
 a0 
b 
a 
 ~m
 ~m
2 m2  1
2m2  1
2 m 1
ek  {1 (i k )  (i k )
}
}
 a2 m (i ) 2 m )
  H k {1 (i k )  (i k )
  H k (a2 m 1 (i )
 
 
b2 m  2 
a2 m  2 
When L such equations are combined:
{E}L1  [P]L2m{b}2m1  [T ]L( 2m1){a}( 2m1)1 {W}L1
Solution will be found, by minimizing the error function J
J  {E }T {E}
This leads to:


{G}  Re[ P ] {W };


[Y ]  Re [ P* ]T [ P] ; [X ]  Re [ P* ]T [T ] ;
* T



[Z ]  Re [T * ]T [T ]

{F}   Re [T * ]T {W }
Rational Fraction Curve Fits
SETS UP EQUATION OF FORM:
 [Y ]
[ X ]T

[ X ]
{b}2 m  21  {G}

 

[ Z ]  L( 4 m  2) {a}2 m1  {F } L1
Y , X , Z , G, F
CONTAIN:  k VALUES
~
H k VALUES
EQUATIONS ARE OVERDETERMINED L  (m  n)
Rational Fraction Approach
- CURVE - FIT FORMULA TO MEASURED DATA
TO FIND (REAL) COEFFICIENTS
a 0 , a1 , ... b0 , b1 , ... bn
- THEN, SOLVE THE TWO POLYNOMIALS TO
DETERMINE EQUIVALENT MODAL PARAMETERS:
..... Ar , r , r ....
-Measuring difference between original and regenerated FRFs using
the derived modal properties.
-Measuring consistency of the various modal parameters for different
model order choice and eliminating those which vary from run to run.
Example
Caution
MDOF Curve-fits: Light Damping
-It is found that some structures are
very well respond to the above
modal analysis procedures.
-This is mainly due to the difficulties
in acquiring good measurements
near resonances.
-This problem is in lightly-damped
structures.
-In such structures, the damping is
not very important, and the structure
is modeled as an undamped one.
-The aim is to find natural frequencies and modal constants only
by using data measured away from the resonance regions.
MDOF Curve-fits: Light Damping
N
 jk () =

sA jk
 2s   2
s=1
 jk (1) =

12
 
1
2
 1
1
  jk (1)  
2
2
  ( )   1  1
1
 jk 1  
2
2
1   2







 

1

  2
2
 jk ( N )   1   N

1
2
 2  1
2
 

 1 A jk 
 A 
 2 jk 


  
  
 1 A jk 

  

A
 2 jk 

 
  

  

1

2
2
2  N
   N A jk 














1
2
2
 2  1
1
2
2
 2  2
MDOF Curve-fits: Light Damping
 jk()   R A jk 
A jk    R1  jk()
1- Measure FRF over frequency range of interest.
2- Locate the resonances and find the corresponding
natural frequencies.
3- Select some data points away from the resonances.
(No. of Points=No. of Modes+2)
4- Using the selected data and compute the modal constants.
5- Construct a regenerated curve and compare with the
measured FRFs.
Selection of Response Data for Identification
1- Complete Modal Presentation
Measured and Regenerated
FRFs
Global frequency Methods in the
Frequency domain (Multiple Curve Fitting)
N
H jk = 
r =1
r
A jk
2r  2  i  r 2r
 N

r A jk

H jk =    2

2
2 
jk
j
k  r = 1 r    i  r r 


  r A jk 


N
j
k


(

)
=


2
2
2




i


jk
r =1
r
r r
()

SO, CAN USE CURVE-FITTING OF
TO FIND
jk
ESTIMATES OF  r &  r FROM SET OF FRFs.
SDOF and MDOF Testing and
Analysis
MODAL TESTING
ANALYSIS
- DIFFERENT VALUES
FOR r; r (MUST
AVERAGE)
+ SINGLE VALUE
(AVERAGED)
FOR r; r
+ SINGLE VALUE
FOR r; r {}r
- MUST REPEAT
FOR/ ALL FRFs
+ SINGLE VALUE
FOR {}r
+ MULTI-VARIATE
MODE INDICATOR
+ MODE INDICATOR
FUNCTION
+ DOUBLE ROOTS
- CANNOT DETECT
DOUBLE MODES
- CONSISTENT DATA
- EXPENSIVE
Modal Analysis Strategies




H




X
X
X


X
X
X

X
2r


 


X
ROW/COL FRFs
(i.e. n FRFs)
 
ROW/COLS FRFs
(i.e. n  p FRFs)
MULTIPLE
ESTIMATES
(i.e. n)
SINGLE
ESTIMATES
r; r;
{}r
MULTIPLE
ESTIMATES
(i.e. n  p)
MULTIPLE
ESTIMATES
(i.e. p)
ONE FRF
SINGLE
ESTIMATES
*
r; r;
*
{}r
r ; r ; rAjk *
Mode Indicator Functions
HOW TO IDENTIFY ‘GENUINE’ MODES?
HOW TO DETECT ‘REPEATED’ MODES?
HOW TO ESTIMATE MODAL FORCING?
ORDINARY MODE INDICATOR
FUNCTION
(FROM ONE ROW/COLUMN OF [H] )
MULTIVARIATE MODE INDICATOR
FUNCTION
(FROM SEVERAL ROWS/COLUMNS OF [H] )
Ordinary Mode Indicator Functions
MIF() 
 Re(H (). H ()
ij
i 1, N
ij
 ( H ()
i 1, N
ij
2
)
Multivariate Mode Indicator
GIVEN:




H 




X
X
X


X
X
X

X




X
P COLUMNS
CMIF
RMIF
MMIF
N ROWS
Global frequency Methods
Global frequency Methods
Global frequency Methods
Global frequency Methods in the
Frequency domain (Multiple Curve Fitting)
N
H jk = 
r =1
r
A jk
2r  2  i  r 2r
 N

r A jk

H jk =    2

2
2 
jk
j
k  r = 1 r    i  r r 


  r A jk 


N
j
k


(

)
=


2
2
2




i


jk
r =1
r
r r
()

SO, CAN USE CURVE-FITTING OF
TO FIND
jk
ESTIMATES OF  r &  r FROM SET OF FRFs.
1-Global Rational Fraction
Polynomial Method (GRFP)
1- The basic of FRP was described for single FRF.
2- The method can be applied to multi-FRF data.
3- The fact is if we take several FRFs from the same
structure, then the denominator will be the same
for all FRFs.
4- For one FRF we had 2(2m+1) unknowns. If we analyze
N FRFs separately, then we have to calculate 2N(2m+1)
unknowns.
5- The number of coefficient for GRFP method is
(N+1)(2m-1)
Global SVD Method
 r  j r k r j r k 

 jk ()   


i  sr 
r 1  i  s r
N
(1)
where
s r   r r  ir 1  2r
 r  j r k 
  R jk ()
 jk ()   
r 1  i  s r 
N
 r  j r k  
  R jk ( )
i jk ( )   jk ( )   sr 
r 1
 i  sr 
N
Global SVD Method
 jk ( ) {1 i
 jk ( ) {1 i
2
2
i 
i 
N
N
0 
i  s1


i


s
2

i }





0
i


s
N

1
 1i 
  
2 i 
   R( )
 

 N i 

0 
i  s1
0 
 s1

i


s
2


i } 



 0
s N  

0
i


s
N

1
 1 i 
  
2 i  
   R( )
 

 N i 

Global SVD Method
Let’s consider a column of FRFs (p FRFs), then:
 1k ( ) 
 ( ) 
()k   2 k    (i  sr )1 k  R k ()
N 1
P1
   P N N  N
 Pk ( ) 
where
P1
 R 1k ( ) 
 R ( ) 
R k ()   2 k 
  
 R Pk ( ) 
 1 k 
  
k    2 k 
  
 N k 
Global SVD Method
IF
gk ()  (i  sr ) k 
1
N1
THEN
ALSO
NN
N1
()k  gk () R k ()
 ()k  Sr gk () Rk ()
NOW TAKE TWO NEARBY FREQUENCIES:
 (i )k  [][g (i )]  [ R(i )]
 (ic )k  [][g (ic )]  [ R(ic )]
Global SVD Method
 (i )k   (ic )k  []([g (i )]  [ g (ic )])  [R(i )]  [R(ic )]
Assume that the effect of out of range modes is constant over
the frequency range.
 (i )k   (ic )k  []([g (i )]  [ g(ic )])
or
 (i )k  [][g(i )]
In a same way
 (i )k  [][sr ][g(i )]
THESE EXPRESSIONS RELATE TO THE RECEPTANCE & MOBILITY
TERMS FOR ONE i .
Global SVD Method
NOW TAKE SEVERAL (L) FREQUENCIES I=1,2,3,…..L
 k   gk   [gk ]  [] [ k ]
PL
P N
N L
k   sr gk   gk   sr   k 
1
ELIMINATE
k  
T
T

g k 
T
T

sr   k  
THIS LEADS TO AN EIGENPROBLEM:
 
T
k

 sr k  zr  0 (4)
WHERE
T
z  
T
Global SVD Method
Matrix [ ] is calculated directly from measured FRFs.
[ ]  [ (i )]  [ (i c )]
The mobility matrix as:
[ ]  i[ (i )]  i[ (i c )]
Global SVD Method
SOLVE 4 USING SVD IN ORDER TO DETERMINE RANK OF
[], [’] AND THUS THE CORRECT NUMBER OF MODES (n)
 Sr ; r=1,2,….n
NEXT TO FIND MODE SHAPES, RETURN TO 2
 r A jk 
  R i ()
 jk ()   
r 1  i  s r 
n
AND FIND r A jk FROM
  jk ( 1 )   (i1  s1 ) 1
  ( )  
1
 jk 2   (i2  s1 )





 

1



(

)
(
i


s
)
jk
L

  L 1
L1
(i1  s2 ) 1
(i2  s2 ) 1

Ln
  1 A jk 
 A 
2
jk 



  
1  
(iL  s n )   n A jk 



n1
Example