1. Modal Parameter Extraction Methods
Modal Analysis and Testing
S. Ziaei-Rad

2. Type of Modal Analysis In this course By domain By Frequency range
Frequency domain (FRFs)
Time domain (IRFs or response history)
By Frequency range
SDOF method
MDOF method
In this course
Single-FRF methods
Multi-FRF methods

3. Preliminary Checks of FRF Data
Visual checks
Low-frequency asymptotes
High-frequency asymptotes
Incidence of anti-resonances
Overall shape of FRF skeleton
Nyquist plot inspection

4. Basic Skeleton Theory
IF IS ASYMPTOTIC TO
IS ALSO ASYMPTOTIC TO ?

5. Mobility Skeleton

6. Skeleton Geometry

7. Skeleton Geometry Mass-dominated characteristics
Stiffness-dominated characteristics
Abnormal characteristics

8. Assessment of Multiple-FRF Data
Principle Response Function (PRF)

9. 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 matrix where:
n number of measurement DOFs
p number of excitation or reference DOFs

10. Complex Mode Indicator Function (CMIF)
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.

11. Other MIFs
MMIF: *
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:
In this version, natural frequencies are identified by zero crossing of
the RMIF values.

12. MIFs Complex Mode Indicator Multivariate Mode Indicator
Function (CMIF)
Multivariate Mode Indicator
Function (MMIF)

13. Modal Analysis Method Curve Fit Analysis: 1- SDOF Methods
2- MDOF Methods

14. BEST ESTIMATES FOR THE MODAL PARAMETERS
Modal Analysis
GIVEN:
MEASURED
FRF DATA:
MODEL:
DETERMINE:
BEST ESTIMATES FOR THE MODAL PARAMETERS

15. SDOF Curve-fit Method Im Re

16. SDOF Modal Analysis
(1)
(2)
(3)

17. Complete FRF

18. 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 .
2- The local maximum value of the FRF is noted and the
frequency bandwidth of the function for a response level
is determined ( ). The two points are thus identified as
(Half-power point)
3- The damping of the mode can now be estimated from one of the
following formulae.
4- The modal constant can be found from:

19. Peak Amplitude Method

20. Peak Amplitude Method
CASE (a)
CASE (b)

21. 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.

22. 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.

23. Properties of Modal Circle
The effect of modal constant
is to scale and rotate the circle

24. Properties of Modal Circle
Consider two points

25. 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

26. 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

27. 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. 12 13 15 14

28. Circle-Fit Analysis Procedure (Step 3)
Locate natural frequency, obtain damping estimate
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
The frequency of maximum response
The frequency of maximum imaginary receptance
The frequency of zero real receptance 12 13 15
(ii) 14
(i)
(iii)

29. Circle-Fit Analysis Procedure (Step 3) Estimation of Natural Frequency

30. Circle-Fit Analysis Procedure (Step 4) (Damping Estimate)
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) 12 13 15 14

31. Circle-Fit Analysis Procedure (Step 4) (Damping Estimate)b c d e
a- linear data
b- random noise
c- error in the data
d- modal analysis error
e- non-linearity

32. Inverse or Line-fit Method
Standard FRF plot format
Inverse FRF plot format

33. SDOF Modal Analysis Using Inverse FRF Data
GENERAL SDOF ASSUMPTION:
RESIDUAL EFFECTS OF OTHER MODES
DEFINE:
ONE OF THE VALUES OF NEAR
AND
AN ‘INVERSE’ FRF PARAMETER

34. SDOF Modal Analysis Using Inverse FRF Data
WHERE

35. Analysis Step One From measured FRF near , fix one point ( at ) and
Calculate for all other points. Plot and fit:
Slopes of best-fit lines for

36. Analysis Step One
Note
Where So

37. ANALYSIS STEP TWO - Plot - Repeat step one for all values of (Compute
- Fit best straight line
- Find
Hence
From Plot:step One
SLOPE =
SLOPE =
INTERCEPT =
INTERCEPT =

38. Line Fit Modal Analysis
Plot of real and imaginary
Line fit modal analysis
a- Plot of Real and Imaginary
b- Slope from a

39. Regenerated FRFs Measured and regenerated without Residual effect
residuals

40. Residuals
LOW-FREQUENCY MODES
HIGH-FREQUENCY MODES
RESIDUALS

41. Representation of Residuals as Linear Functions

42. High Frequency Residual
Residuals
Low Frequency
Residual
(L.F. Residual)
High Frequency Residual
(H.F. Residual)

43. L.F. Residuals (Rigid Body Modes)Z

44. ALL TERMS +VE ADDITIVE
H.F. Residuals
ALL TERMS +VE ADDITIVE
SOME TERMS +VE, SOME -VE
 TENDENCY TO CANCEL

45. Modal Analysis Methods

46. Modal Analysis in Frequency Domain

47. MDOF Curve-fit Method H

48. Curve - Fitting In General (Nonlinear Least-Squares)
MEASURED FRF DATA:
THEORETICAL MODEL FOR FRF DATA:

49. Modal Analysis Using Rational Fractions
USE ALTERNATIVE FORMAT FOR FRF:
INSTEAD OF PARTIAL FRACTION

50. Rational Fraction Curve Fits
LET
AND
GIVEN SEVERAL VALUES OF
FIND TO MINIMISE

51. Rational Fraction Curve Fits
When L such equations are combined:
Solution will be found, by minimizing the error function J
This leads to:

52. Rational Fraction Curve Fits
SETS UP EQUATION OF FORM:
CONTAIN: VALUES
VALUES
EQUATIONS ARE OVERDETERMINED

53. Rational Fraction Approach
- CURVE - FIT FORMULA TO MEASURED DATA TO FIND (REAL) COEFFICIENTS
- THEN, SOLVE THE TWO POLYNOMIALS TO DETERMINE EQUIVALENT MODAL PARAMETERS:
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.

54. Example

55. Caution

56. 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.

57. MDOF Curve-fits: Light Damping

58. MDOF Curve-fits: Light Damping
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.

59. Selection of Response Data for Identification 1- Complete Modal Presentation

60. Measured and Regenerated FRFs

61. SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.
Global frequency Methods in the Frequency domain (Multiple Curve Fitting)
SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.

62. 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
+ SINGLE VALUE FOR {}r
+ MULTI-VARIATE MODE INDICATOR
- MUST REPEAT FOR/ ALL FRFs
+ MODE INDICATOR FUNCTION
+ DOUBLE ROOTS
- CANNOT DETECT DOUBLE MODES
- CONSISTENT DATA
- EXPENSIVE

63. Modal Analysis Strategies
MULTIPLE ESTIMATES
ROW/COL FRFs
r; r; *
(i.e. n FRFs)
(i.e. n)
{}r
SINGLE ESTIMATES
MULTIPLE ESTIMATES
ROW/COLS FRFs
r; r;
(i.e. n  p FRFs) *
(i.e. n  p)
{}r
MULTIPLE ESTIMATES
(i.e. p)
SINGLE ESTIMATES
r ; r ; rAjk *
ONE FRF

64. 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] )

65. Ordinary Mode Indicator Functions

66. Multivariate Mode Indicator
CMIF
RMIF
MMIF
GIVEN:
N ROWS
P COLUMNS

67. Global frequency Methods

68. Global frequency Methods

69. Global frequency Methods

70. SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.
Global frequency Methods in the Frequency domain (Multiple Curve Fitting)
SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.

71. 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)

72. Global SVD Method
(1)
where

73. Global SVD Method

74. Global SVD Method Let’s consider a column of FRFs (p FRFs), then:
where

75. NOW TAKE TWO NEARBY FREQUENCIES:
Global SVD Method IF
THEN
ALSO
NOW TAKE TWO NEARBY FREQUENCIES:

76. Global SVD Method
Assume that the effect of out of range modes is constant over
the frequency range. or
In a same way
THESE EXPRESSIONS RELATE TO THE RECEPTANCE & MOBILITY
TERMS FOR ONE i .

77. Global SVD Method (4) NOW TAKE SEVERAL (L) FREQUENCIES I=1,2,3,…..L
ELIMINATE
THIS LEADS TO AN EIGENPROBLEM:
(4)
WHERE

78. Global SVD Method Matrix is calculated directly from measured FRFs.
The mobility matrix as:

79. NEXT TO FIND MODE SHAPES, RETURN TO 2
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
AND FIND FROM

80. Example