Characteristics and Presentation
of MDOF FRF Data
Modal Analysis and Modal Testing
S. Ziaei Rad
1
Receptance and Impedance FRF
Parameters
The Relation between different form of FRF can be stated as before:
[Y ( )]  i[ H ( )]
[ A( )]  i[Y ( )]
[ A( )]   [ H ( )]
2
A general element of the receptance is given by:
H jk ( ) 
2
Xj
Fk
, Fl  0
l  1,, N (l  k )
Receptance and Impedance FRF
Parameters
Now, let’s look at the Impedance matrix [Z]:
{X }  [ H ]{F}
1
{F}  [Z ]{F}  [ H ] {F}
Therefore, we can not simply write:
1
H jk ( ) 
Z jk
Looking at the definition of a typical element of [Z]:
Z jk ( ) 
3
Fj
Xk
, Xl  0
l  1,, N (l  k )
Receptance and Impedance FRF
Parameters
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4
To measure the receptance, we should make sure that
just a single excitation force is applying on the
structure.
To measure an impedance property all DOFs except
one should be grounded.
Such a condition is almost impossible to achieve in
practical situation.
Therefore, only types of FRF which can expect to
measure directly are those of the mobility or
receptance type.
Some Definitions
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5
A Point Mobility (or receptance) is one where the
response DOF and the excitation coordinate are
identical.
A Transfer Mobility is one where the response and
excitation DOFs are different.
A Direct Mobility is one where types of DOFs for
response and excitation are identical. (both in x)
A Cross Mobility is one where types of DOFs for
response and excitation are not identical. (one in x and
other in y direction)
FRF Plot in MDOF System
Typical mobility FRF plot for MDOF system
(individual modal contribution)
6
Point and Transfer FRFs
Point FRF
7
Transfer FRF
Point and Transfer FRFs
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8
There is an anti-resonance after each resonance in
point FRF.
In point FRF the modal constant for every mode is
positive, it being the square of a number.
In transfer FRF, there is an anti-resonance or a minima
after each resonance.
We expect a transfer FRF between two positions
widely separated on the structure to exhibit fewer antiresonances than one for two points relatively close
together. (the further apart are the two points, the
more likely are the two eigen vectors elements to
alternate in sign as one progress through the modes.
Ponit and transfer FRF for 6DOF
system
k1
m1
x1
k3
k2
m2
x2
m3
x3
k4
m4
x4
m5
x5
m1=m2=m3=m4=m5=m6=1 Kg
k1=k2=k3=k4=k5=k6=100000 N/m
9
k6
k5
m6
x6
FRFs of 6DOF System
10
H11
H21
H31
H41
H51
H61
Display of FRF Data For Damped
Systems
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Bode Plots
Nyquist diagrams
Real and Imaginary plots
Three-dimensional plots
2DOF System
k2
k1
m1
x1
m2
x2
1  0.02
m1=m2=1 Kg
Hysteretic damping
k1=k2=360 kN/M
2  0.04
 jrkr
H jk ( )  2
 r   2  ir2
12
Bode Plot
H11
13
H12
Nyquist Plot
H11
14
H12
Real and Imaginary
H11
15
H12
3D Plot
16
Cr exp j r 
1
2

1     jr
 r 
Cr exp j r 
2

1     j r
 r 
17
 Dr
2

1     j r
 r 

r
Cr exp j r 
2

1     j r
 r 
18
Conclusions
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19
The purpose of this session has been to
predict the form which will be taken by plots of
FRF data using the different display format.
Although the graphs were taken from some
theoretical models, they can help to
understand and interpret actual measured
data.