The spectral decomposition associated to complex modes, leads to a
representation of the transfer function as a sum of modal contributions
[(s)] =
2N
j=1
æ
ç
ç
è
{cj}{jTb}
s-j
ö
÷
÷
ø
=
2N
j=1
æ
ç
ç
è
[Rj]
s-j
ö
÷
÷
ø
(2.21)
For applications in identification from experimental data, one can
only determine modes whose poles are located in the test frequency range. The full
series thus need to be truncated. The contributions of out-of-band
modes cannot be neglected for applications in structural dynamics.
One thus introduces a high frequency residual correction for truncated
high frequency terms and, when needed, (quite often for suspended test
articles) a low frequency residual for modes below the measurement
frequency band.
These corrections depend on the type of transfer
function so that the SDT uses the IDopt variable (see the
reference section on the idopt function) to define the current type. IDopt.Residual specifies which corrections are needed (the default is 3
which includes both a low and high frequency residuals). IDopt.Data specifies if the FRF is force to displacement, velocity or
acceleration.
For a force to displacement transfer function with low and high frequency correction), the pole/residue model (also called partial fraction expansion) thus takes the form
[(s)] =
j Î identified
æ
ç
ç
ç
ç
ç
è
[Rj]
s-j
+
[
-
R
j
]
s-
-
j
ö
÷
÷
÷
÷
÷
ø
+ [E] +
[F]
s2
(2.22)
The SDT always stores pole/residue models in the
displacement/force format. The expression of the force to acceleration
transfer function is thus
[A(s)] =
j Î identified
æ
ç
ç
ç
ç
ç
è
s2[Rj]
s-j
+
s2[
-
R
j
]
s-
-
j
ö
÷
÷
÷
÷
÷
ø
+ s2[E] + [F]
(2.23)
The nominal pole/residue model above is used when IDopt.Fit='Complex'. This model assumes that complex poles come in conjugate pairs and that the residue matrices are also conjugate which is true for real system.
The complex residues with asymmetric pole structure
(IDopt.Fit='Posit') only keep the poles with positive imaginary parts
[(s)] =
j Î identified
æ
ç
ç
è
[Rj]
s-j
ö
÷
÷
ø
+ [E] +
[F]
s2
(2.24)
which allows slightly faster computations when using id_rc for the identification but not so much so that the symmetric pole pattern should not be used in general. This option is only maintained for backward compatibility reasons.
The normal mode residues with symmetric pole structure (IDopt.Fit='Nor')
[(s)] =
j Î identified
æ
ç
ç
è
[Tj]
s2+2jjs+j2
ö
÷
÷
ø
+ [E] +
[F]
s2
(2.25)
can be used to identify normal modes directly under the assumption of
modal damping (see damp ).
Further characterization of the properties of a given pole/residue model is given by a structure detailed under the xfopt Shapes at DOF section.
The residue matrices res are stored using one
row for each pole or asymptotic correction term and, as for FRFs (see the xf format), a column for each SISO
transfer function (stacking NS columns for actuator 1, then NS
columns for actuator 2, etc.).
res = [
·
·
·
...
...
...
Rj(11)
Rj(21)
...
Rj(12)
Rj(22)
...
·
·
·
·
·
·
·
·
·
·
·
·
E11
E21
...
E12
E22
...
F11
F21
...
F12
F22
...
]
(2.26)
The normal mode residues (IDopt.Fit='Normal') are stored in a
similar fashion with for only difference that the Tj are real while
the Rj are complex.
PDF
(s)] = 
j}{
j
j
js+