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## psi2nor

Purpose

Estimation of normal modes from a set of scaled complex modes.

Syntax

```[wj,ga,cps,pbs] = psi2nor(po,cp)
[wj,ga,cps,pbs] = psi2nor(po,cp,ncol,NoCommentFlag)
```

Description

psi2nor should generally be used through id_nor. For cases with as many and more sensors than modes, psi2nor gives, as proposed in Ref. , a proper approximation of the complex mode outputs cp=[c] [ψ] (obtained using id_rm), and uses the then exact transformation from complex to normal modes to define the normal mode properties (modal frequencies wj, non-proportional damping matrix ga, input pbs=[φ]T[b] and output cps=[c] [φ] matrices).

The argument ncol allows the user to specify the numbers of a restricted set of outputs taken to have a collocated input (pbs=cps(ncol,:)').

If used with less than four arguments (not using the NoCommentFlag input argument), psi2nor will display two indications of the amount of approximation introduced by using the proper complex modes. For the complex mode matrix ψT (of dimensions NT by 2NT because of complex conjugate modes), the properness condition is given by ψTψTT=0. In general, identified modes do not verify this condition and the norm ||ψTψTT || is displayed

``` The norm of psi*psi' is 3.416e-03 instead of 0
```

and for well identified modes this norm should be small (10−3 for example). The algorithm in psi2nor computes a modification Δψ so that ψPTT+Δψ verifies the properness condition ψPTψPTT=0 . The mean and maximal values of abs(dpsi./psi) are displayed as an indication of how large a modification was introduced

``` The changes implied by the use of proper cplx modes are
0.502 maximum and 0.122 on average
```

The modified modes do not necessarily correspond to a positive-definite mass matrix. If such is not the case, the modal damping matrix cannot be determined and this results in an error. Quite often, a non-positive-definite mass matrix corresponds to a scaling error in the complex modeshapes and one should verify that the identification process (identification of the complex mode residues with id_rc and determination of scaled complex mode outputs with id_rm) has been accurately done.

Warnings

The complex modal input is assumed to be properly scaled with reciprocity constraints (see id_rm). After the transformation the normal mode input/output matrices verify the same reciprocity constraints. This guarantees in particular that they correspond to mass-normalized analytical normal modes.

For lightly damped structures, the average phase of this complex modal output should be close to the −45o line (a warning is given if this is not true). In particular a sign change between collocated inputs and outputs leads to complex modal outputs on the +45o line.

Collocated force to displacement transfer functions have phase between 0 and −180o, if this is not verified in the data, one cannot expect the scaling of id_rm to be appropriate and should not use psi2nor.   