modal parameter extraction methods modal analysis and testing s. ziaei-rad
TRANSCRIPT
Modal Parameter Extraction Methods
Modal Analysis and Testing
S. Ziaei-Rad
Type of Modal Analysis
By domain1. Frequency domain (FRFs)2. Time domain (IRFs or response history)
By Frequency range1. SDOF method2. MDOF method
In this course1. Single-FRF methods2. Multi-FRF methods
Preliminary Checks of FRF Data
Visual checks
1. Low-frequency asymptotes
2. High-frequency asymptotes
3. Incidence of anti-resonances
4. Overall shape of FRF skeleton
5. Nyquist plot inspection
Basic Skeleton Theory
IS ALSO ASYMPTOTIC TO ?2M Y
IF IS ASYMPTOTIC TO 1M Y
Mobility Skeleton
Skeleton Geometry
A
SlogDCDEBD
2A
2SlogDCECBD2
1
21
Smm
S1
m
mmlog
Y
Ymlog
21
1
21
1
22s
m
mm
k
m
2Rm
1m1
12s 21
k
Skeleton Geometry
Mass-dominated characteristics
Stiffness-dominated characteristics
Abnormal characteristics
Assessment of Multiple-FRF Data
])}( { )}( {)}([{][ 1112111 LnpLLnpL HHHA
TnpnpnpLLLnpL VUA ][][][][
npLLLnpL UP ][][][ 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 typicallyconsists of an matrix where: n number of measurement DOFs p number of excitation or reference DOFs
pn
Complex Mode Indicator Function (CMIF)
Hpppnnnpn VUH )]([)]([)]([)]([
pnT
pp npCMIF
)]([)]([)]([
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:
}F]{H[]H[}F{]H[]H[]H[]H[ RT
RIT
IRT
R
results from the eigenvalue solution equation (*) for each frequencyAnd 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 ofMaximum values for the CMIF.
*
RMIF:
}{}{][][ FFHH RI
In this version, natural frequencies are identified by zero crossing of the RMIF values.
MIFs
Complex Mode Indicator Function (CMIF)
Multivariate Mode IndicatorFunction (MMIF)
Modal Analysis Method
Ar
H jkH jk
H
Curve Fit Analysis:1- SDOF Methods2- MDOF Methods
Modal Analysis
GIVEN:
DETERMINE:
MEASURED
FRF DATA:
MODEL:
N
1 = r r22
r i+ rA
H )(
BEST ESTIMATES FOR THE MODAL PARAMETERS
222
111
A
A
H(w)
SDOF Curve-fit Method
1 1 1, , A
2 2 2, , A
3 3 3, , A
H jk
H jk
H jk Im
Im
ImRe
ReRe
SDOF Modal Analysis
jk
s jk
s s ss =
N
( ) = A
i 2 2 2
1
jk
r jk
r r r
s jk
s s ss = r
N
( ) = A
i
A
i
2 2 2 2 2 21
jk
r jk
r r rr jk( )
A
iR
2 2 2
r
(1)
(2)
(3)
Complete FRF
Peak Amplitude Method1- 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:
rH
2/H
rrrr
bar
2,
2 2
22
rrrrr
r HAorA
H
22
|ˆ| |ˆ|
a b,
Peak Amplitude Method
ra b
r
2
b a 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 MethodProperties 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 Re
r
tan(1 ( ) )
tan (1 ( ) ) /
r2
2r
r
r
(1 tan )
1+ (1 ( ) )
= 2
= 0,
2r2
r
r2
r2
r2
r2 r
r2
r
2
r r
2
2
r
2
2
rrr i
22 )/(1
1)( The effect of modal constant
is to scale and rotate the circle
Properties of Modal Circle
b
Im
Re
r
tan
tan
br b
2
r r2
aa r
2
r r2
2
2
r
122
r
21
22
r
21
b21
a21
r
bar
b21
a212
r
2b
2a
r
2
,
tan+tan
2
tan+tan
1
POINTS; POWERHALF
b
a
a
b
Consider two points
90 ba
rr
jkr
jkr
AD
2
Circle-Fit Analysis Procedure
1- Select point to be used2- Fit circle, calculate quality of fit3- Locate natural frequency, obtain damping estimate4- Calculate multiple damping estimates and scatter5- Determine modal constant modulus and argument
Circle-Fit Analysis Procedure (Step 1)
SELECT DATA POINTS
Im( )
Re( )
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
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( )
Re( )
r
1
23
1514
13
12
Im( )
Re( )
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
1.The frequency of maximum response2.The frequency of maximum imaginary receptance3.The frequency of zero real receptance
Im( )
Re( )
1514
13
12
(i)(ii)
(iii)
Circle-Fit Analysis Procedure (Step 3)Estimation of Natural Frequency
Circle-Fit Analysis Procedure (Step 4)(Damping Estimate)
Im( )
Re( )
15 14
13
12ab
a
b
r
ra b
r a b
1
tan( / 2) + tan( / 2)
2 2
2
-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)
Circle-Fit Analysis Procedure (Step 4)(Damping Estimate)
a b
cd e
a- linear datab- random noisec- error in the datad- modal analysis errore- non-linearity
Inverse or Line-fit Method2 2 22 2 2
2
2 ) d ( ) m -k (
d i
) d ( ) m -k (
m -k =
d i + m -k
1 = )(
) d (i + ) m - k ( = )( 2-1 )c(i Or
Standard FRF plot format Inverse FRF plot format
SDOF Modal Analysis Using Inverse FRF Data
GENERAL SDOF ASSUMPTION: 222 r
rrr
r R i
A= )(
r
' ( ) = ( ) ( )
( ) =
( )
'
2 2
AND
DEFINE:
RESIDUAL EFFECTS OF OTHER MODES
rONE OF THE VALUES OF NEAR
AN ‘INVERSE’ FRF PARAMETER
)Im()Re(
222222
iA
ii)(
r
rrrrrr
SDOF Modal Analysis Using Inverse FRF Data
( )
i i
Ar r r r r r
r
2 2 2 2 2 2
,
RE( ) i IM( )
RE( ) m c IM( ) m c
m a b
m b a A a i b
R R I I
R r r r r r
I r r r r r r r r
2 2
2 2 2
2 2 2
WHERE
Analysis Step One
RE( ) IM( )
22
r j
From measured FRF near , fix one point ( at ) and
Calculate for all other points. Plot and fit:
Slopes of best-fit lines for
RE vs m
IM vs m
R j
l j
2
2
Analysis Step One
m n d
m n d
R R R
l l l
2
2
n a d b a
n b d b a
R r R r r r r r
I r l r r r r r
;
;
2 2
2 2
r R r R
r
r r r R
r r
d (p ) n
(q p) ( pq)
a (p ) ( p ) d
b p a
2
2 2
1
1
1 1
/
/
/
.
pn
n
qd
d
l
R
l
R
Note
Where
So
ANALYSIS STEP TWO- Repeat step one for all values of j
(m ) (m ) (m ) (m ) )R R l l1 2 1 2, , ... , , , ...
From Plot:step One
(m ) vs (m ) vsR l , 2 2
n dR R 2
n dl l 2 n ; d ; n ; dR R l l
r r r r; ; a ; b
jth
(Compute
- Plot
- Fit best straight line
- FindHence
mR
mR j2
jnR
dR
Ml
nldl
2
SLOPE =INTERCEPT =
SLOPE =INTERCEPT =
Line Fit Modal Analysis
Line fit modal analysisPlot of real and imaginary
Line fit modal analysisa- Plot of Real and Imaginaryb- Slope from a
Regenerated FRFs ~
~
~ ~ ~
jk
r jk
r r rr = m
m
( ) = A
i2 2 21
2
Measured and regenerated without Residual effect
Measured and regenerated withresiduals
Residuals
jkr = m
mr jk
r r r
r jk
r r rr = 1
m
r jk
r r rr = m
N
( ) = A
i
A
i
A
i
1
2
1
2
2 2 2
2 2 2
1
2 2 21
jk
r jk
r r r jkR
r = m
m
jkR( )
A
i M K
2 2 2 21 1
1
2
LOW-FREQUENCY MODES
HIGH-FREQUENCY MODES
RESIDUALS
Representation of Residuals as Linear Functions
Residuals
HL2
m
mr r2r
22r
r RR1
i
A)(H
2
1
1m
1rr2
1m
1r22
r
rL
11
A1A
R
2
r
rorN
1mr22
r
rH
AAR
2
Low Frequency Residual
(L.F. Residual)
High Frequency Residual(H.F. Residual)
L.F. Residuals(Rigid Body Modes)
fxM a fdI a
aa dxz fI
d
M
1
a
2
a
2
2 I
d
M
11
f
z
LR
gIM ,
aax ,
f
Zd
H.F. Residuals
ALL TERMS +VE ADDITIVE
SOME TERMS +VE, SOME -VE
TENDENCY TO CANCEL
:kj
:kj
N
1nr2r
krjrjkR
Modal Analysis Methods
Modal Analysis in Frequency Domain
MDOF Curve-fit Method
NNN A
A
A
222
111
H
Curve - Fitting In General(Nonlinear Least-Squares)
jkm( ) = 1 1
jk ( ) = 1 1
p
eee
M
RR
M
Ms sss
s
EWEE
MK i
A
1
2
11
111
21
221
21
;
1
1
2
1
E
qq A A etc ; , , .... , 0 1 2
MEASURED FRF DATA:
THEORETICAL MODEL FOR FRF DATA:
Modal Analysis Using Rational Fractions
USE ALTERNATIVE FORMAT FOR FRF:
NN
NN
aiaa
ibibb)H(
2210
)1(21210
)(
)( )(
INSTEAD OF PARTIAL FRACTION
iS
A or
i 2
A )H(
r
r
rr22
r
r
Rational Fraction Curve Fits
~
)(
)(
10
10 mkk H
iaa
ibbe
k
)(HH kmm
k ~~
mkk
mk
mkmkk iiaaHibibbe 2
10)1(2
2210 )( )( ~
)( )(
2 kk eE
mkk H
~ ,
a , a , ... bn0 1 kE
LET
AND
GIVEN SEVERAL VALUES OF
FIND TO MINIMISE
Rational Fraction Curve Fits
) )()((~
}) () (1{ ~
}) () (1{ 22
1212
22
1
0
22
22
1
0
22 mm
mm
mk
m
mkk
mk
m
mkkk iaiaH
a
a
a
iiH
b
b
b
iie
When L such equations are combined:
11)12()12(1221 }{}{][}{][}{ LmmLmmLL WaTbPE
Solution will be found, by minimizing the error function J
}{}{ EEJ T
This leads to:
}{][Re} {; }{][Re}{
][][Re]; [][][Re]; [][][Re][**
***
WTFWPG
TTZTPXPPYTT
TTT
Rational Fraction Curve Fits
SETS UP EQUATION OF FORM:
EQUATIONS ARE OVERDETERMINED L (m n)
VALUES
CONTAIN: VALUESk
kH~
FGZXY , , , ,
112
122
)24(}{
}{
}{
}{
][][
][][
Lm
m
mL
T F
G
a
b
ZX
XY
Rational Fraction Approach
- CURVE - FIT FORMULA TO MEASURED DATA TO FIND (REAL) COEFFICIENTS
- THEN, SOLVE THE TWO POLYNOMIALS TO DETERMINE EQUIVALENT MODAL PARAMETERS:
a , a , . . . b , b , . . . b0 1 0 1 n
.... , , A..... rrr
-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
jk
s jk
ss = 1
N
( ) = A
2 2
jk
jk
jk( ) =
A
A 1 1
212 1
22
12 1
1
2
jk
jk
jk N N N
jk
jk
N jk
( )
( )
( )
A
A
A
1
112
12 1
22
12 1
12
22 1
22
22 1
12 2 1
22 2 1
1
2
MDOF Curve-fits: Light Damping
jk jk( ) R A
A R ( )jk jk 1
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)
i
A =H
N
1 =r 2rr
22r
jkrjk
i
A
=)(N
1 =r 2rr
22r
j kjkr
jk
i
A =H
j k
N
1 =r 2rr
22r
jkr
jkjk
SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.
)(jk
r r
SDOF and MDOF Testing and Analysis
MODAL TESTING ANALYSIS
- DIFFERENT VALUES FOR r; r (MUST AVERAGE)
- MUST REPEAT FOR/ ALL FRFs
+ SINGLE VALUE (AVERAGED) FOR r; r
+ SINGLE VALUE FOR r; r {}r
+ SINGLE VALUE FOR {}r
+ MODE INDICATOR FUNCTION
- CANNOT DETECT DOUBLE MODES
+ MULTI-VARIATE MODE INDICATOR
+ DOUBLE ROOTS
- CONSISTENT DATA
- EXPENSIVE
Modal Analysis Strategies
2r
XX
XX
XX
XX
H
ONE FRFSINGLE
ESTIMATESr ; r ; rAjk *
ROW/COLS FRFs
ROW/COL FRFsMULTIPLE
ESTIMATES r; r;
SINGLE ESTIMATES
{}r
MULTIPLE ESTIMATES
MULTIPLE ESTIMATES
(i.e. n FRFs)
(i.e. n p FRFs) (i.e. n
p)
(i.e. p)
(i.e. n) *
r; r;
{}r
*
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
N,1i
2
ij
ijN,1i
ij
))(H(
)(H.)(HRe(
)(MIF
Multivariate Mode Indicator
XX
XX
XX
XX
HGIVEN:
P COLUMNS
N ROWS
CMIF
RMIF
MMIF
Global frequency Methods
Global frequency Methods
Global frequency Methods
Global frequency Methods in the Frequency domain (Multiple Curve Fitting)
i
A =H
N
1 =r 2rr
22r
jkrjk
i
A
=)(N
1 =r 2rr
22r
j kjkr
jk
i
A =H
j k
N
1 =r 2rr
22r
jkr
jkjk
SO, CAN USE CURVE-FITTING OF TO FIND ESTIMATES OF & FROM SET OF FRFs.
)(jk
r r
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
N
1r r
krjr
r
krjrjk sisi
)(
2rrrrr 1is
)(Rsi
)( jk
N
1r r
krjrjk
(1)
)()()(1
jk
N
r r
krjrrjkjk R
sisi
where
Global SVD Method
)(
0
0
} {)( 2
1
1
2
1
21
R
si
si
si
iN
i
i
N
iNiijk
)(
0
0
0
0
} {)( 2
1
1
2
11
21
R
si
si
si
s
s
iN
i
i
NN
iNiijk
Global SVD Method
1P
k1N
k1
NNr
NP
1PPk
k2
k1
k )(R)si(
)(
)(
)(
)(
kN
k2
k1
k
)(R
)(R
)(R
)(R
Pk
k2
k1
k
Let’s consider a column of FRFs (p FRFs), then:
where
Global SVD Method
IF
)(R)(g)( kkk
1N
k1
NNr
1Nk )si()(g
THEN
ALSO
NOW TAKE TWO NEARBY FREQUENCIES:
)()()( kkrk RgS
)]([)](][[)(
)]([)](][[)(
cicikci
iiki
Rg
Rg
Global SVD Method )]([)]([)])([)](]([[)()( ciiciikciki RRgg
Assume that the effect of out of range modes is constant over the frequency range.
)])([)](]([[)()( ciikciki gg
or )](][[)( iki g
In a same way )](][][[)( irki gs
THESE EXPRESSIONS RELATE TO THE RECEPTANCE & MOBILITY
TERMS FOR ONE i .
Global SVD MethodNOW TAKE SEVERAL (L) FREQUENCIES I=1,2,3,…..L
][][][ kkLNk
NPLPk gg
krkkrk sggs 1
kgELIMINATE
TTkr
TTk s
THIS LEADS TO AN EIGENPROBLEM:
0zs rT
krT
k
Tz WHERE
(4)
Global SVD Method
Matrix is calculated directly from measured FRFs. ][
)]([)]([][ cii
)]([)]([][ cii ii
The mobility matrix as:
Global SVD MethodSOLVE 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
)(Rsi
A)( i
n
1r r
jkrjk
jkr AAND FIND FROM
1njkn
jk2
jk1
nL
1nL
11L
122
112
121
111
1LLjk
2jk
1jk
A
A
A
)si()si(
)si()si(
)si()si(
)(
)(
)(
Example