DIRECT UPDTING OF THE FE MODEL
6.2. UPDATING TECHNIQUES
Two significant direct updating techniques are employed in this updating exercise as: (a) Berman and Nagy updating technique (Berman and Nagy 1983) and (b) matrix mixing updating (Friswell and Mottershead 1995). In this study, it has been taken care that the system matrices especially the mass matrix are updated with minimum changes. It may be mentioned that there are other techniques which only update the stiffness matrix without any changes in the original analytical mass matrix, namely, Baruch and Bar-Itzhack updating technique (Baruch and Bar-Itzhack 1978; Baruch 1978), a recently proposed technique – incomplete data handling method (IDHM) (Carvalh et al. 2007) etc. But those techniques have difficulties in reproduction of the mode shapes exactly and hence are considered in this updating study. Outlines of the two techniques considered in this study are briefly mentioned in the following sub-sections.
6.2.1. Berman and Nagy updating technique
Berman and Nagy (1983) technique uses the identified modal parameters in terms of natural frequencies and mode shapes for updating. Firstly, the mass matrix is updated ensuring the orthogonality of the measured modes. The updating of the mass matrix is considered by finding the updated mass matrix [MU] that minimizes a cost function (as in Eq. (6.1)) in terms of the following norm when the experimentally identified mode shape matrix
E
and the analytical mass matrix [MA] are given.
1/2
1/21 A U A A
J M M M M (6.1)
This minimization is subjected to the orthogonality constraint expressed as
E
T MU
E
I (6.2)The difference between the updated and the analytical mass matrices are minimized. The expression of the updated mass matrix is provided as
MU
MA
MA
E
MA1
I MA
MA1
E
T MA
(6.3)where, the matrix MA is defined as MA
E
T MA
E
. Subsequently, the updated stiffness matrix is obtained by minimizing a cost function in terms of the following norm.
1/2
1/22 U U A U
J M K K M (6.4)
Two constraints are imposed on the updated stiffness matrix [KU] such that: (a) [KU] is able to reproduce the measured modal data and (b) at the same time, [KU] is symmetric. These constraints are expressed by Eqs. (6.5) and (6.6) respectively.
KU
E
MU
E
E (6.5)
KU
T
KU
(6.6) where,
E represents the diagonal matrix where the diagonal elements are the measured natural frequencies in eigen-value format. The difference between the updated and the analytical stiffness matrices are minimized as in the case of mass matrix updating. The expression of the updated stiffness matrix is written as
T T
U A A E E U U E E A
T T
U E E A E E U
T
U E E E U
K K K M M K
M K M
M M
(6.7)
Details regarding these derivation can be found in literatures like Friswell and Mottershead (1995), Berman and Nagy (1983) etc. The measured mode shapes vectors need to be same as the size of analytical system matrices. Usually mode shapes are experimentally identified along much lesser DOF locations than the DOF locations available in the corresponding FE
model. In such cases measured mode shape vectors can be expanded or the analytical system matrices can be reduced to the size of a measured mode shape vector.
6.2.2. Matrix mixing updating technique
Berman and Nagy technique obtains the updated mass matrix with an objective of minimal changes with respect to the initial analytical mass matrix. However, still there exists interest to work with an alternative technique which may also perform well. It is known that mass and stiffness matrices can be exactly reconstructed using all the modal frequencies and all mode shapes, when (a) the mode shapes are mass normalized and (b) all DOFs are retained in the mode shapes. Reconstruction of mass and stiffness matrices are carried out as
1
1
T n T
i i
i
M
(6.8)
1
1
2 1n T
T i i
i i
K
(6.9)where,
i is the ith mode shape,
is the matrix of eigen values and i is the ith natural frequency. Usually the measured modal data is incomplete as measurements of responses are carried out generally at limited numbers of coordinates. Moreover the numbers of identified modes are quite fewer than the numbers of analytical DOFs. Such incompleteness in estimation of modal parameters gives raises various problems in estimating of mass and stiffness matrices. Some attempts were carried out to avoid such difficulties by different investigators (Ross 1971; Thoren 1972; Luk 1987). Matrix mixing technique (Caser 1987;Link et al. 1987) is considered as a development over the methods proposed by Ross (1971) and Thoren (1972). Matrix mixing technique demonstrates that if the estimated mode shapes are expanded to the size of analytical system matrices, then computational difficulties can be reduced. Generally, number of measured modes, p, is fewer than the number of modes based
on FE model. Matrix mixing technique uses the data from the FE model to compensate the gaps in measured data. Thus, updated mass and stiffness matrices can be computed as
1
1 1
p n
T T
U Ei Ei Ai Ai
i i p
M
(6.10)
1
2
2
1 1
T T
p n
Ei Ei Ai Ai
U
i Ei i p Ai
K
(6.11)where,
Ei and
Ai denotes experimental and analytical ith mode shape vectors respectively, ωEi and ωAi represents corresponding experimental and analytical natural frequencies respectively. In many cases, number of experimentally identified modes is much less than the number of modes available from the FE model. In such cases, it is wiser to compute the summation terms associated to higher frequencies as in Eqs. (6.10) and (6.11) using the following equations.
1
1 1
n p
T T
Ai Ai A Ai Ai
i p i
M
(6.12)
1
2 2
1 1
T p T
n
Ai Ai Ai Ai
A
i p Ai i Ai
K
(6.13)6.2.3. Expansion of the experimental mode shapes
In the present study, both the Berman-Nagy technique and matrix mixing technique are implemented using the expanded mode shape vectors. The system equivalent reduction expansion process (SEREP) (O’Callahan et al. 1989), an extensively used technique in this regard, is applied for the expansion of the experimental mode shapes. SEREP uses the analytical mode shapes to produce the transformation matrix enabling the expansion process.
The analytical mode shapes are partitioned into the master and slave coordinates as
Am A
As
(6.14)
DOFs corresponding to the measured coordinates are considered as master category, while rest of the DOFs are considered as slave. Finally the transformation matrix is computed as
Am
Am
As
T
(6.15)
where,
Am
is the pseudo inverse of
Am
and computed as
Am
Am
T Am
1
Am
T (6.16) Experimental mode shapes then can be expanded to the size of FE model as
Ef T E
(6.17)