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5.6  Pole/residue models

The spectral decomposition associated to complex modes, leads to a representation of the transfer function as a sum of modal contributions

   [α(s)]  =  
2N
j=1
 


{cψj}{ψjTb}
s−λj



2N
j=1
 


[Rj]
s−λj



    (5.21)

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 ci.IDopt options (see the reference section on the idopt function) to define the current type. ci.IDopt.Residual specifies which corrections are needed (the default is 3 which includes both a low and high frequency residuals). ci.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

    [α(s)]  =  
 
j ∈  identified
 


[Rj]
s−λj
+
[Rj]
sλj



+ [E] + 
[F]
s2
    (5.22)

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 ∈  identified
 


s2[Rj]
s−λj
+
s2[Rj]
sλj



s2[E] + [F]     (5.23)

The nominal pole/residue model above is used when ci.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 (ci.IDopt.Fit='Posit') only keep the poles with positive imaginary parts

  [α(s)]  =  
 
j ∈ identified
 


[Rj]
s−λj



+ [E] + 
[F]
s2
    (5.24)

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 (ci.IDopt.Fit='Nor')

    [α(s)]  =  
 
j ∈ identified
 


[Tj]
s2+2ζjωjsj2



+ [E] + 
[F]
s2
    (5.25)

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 Shapes at DOFs 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)… 
    ⋮  ⋱ 
    E11E21E12E22… 
    F11F21F12F22… 
    
]     (5.26)

The normal mode residues (ci.IDopt.Fit='Normal') are stored in a similar fashion with for only difference that the Tj are real while the Rj are complex.


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