Model reduction
4.6 Structural damage detection using incomplete modal data
4.6.1 M o d e s h a p e e x p a n s i o n
The measured modes are incomplete in practice because of the limited number of sensors, and because the rotational DOFs are usually difficult to measure. Law et al.
(1998) have presented a mode shape expansion by which the mode shape of a limited number of measured DOFs is expanded to the full dimension of the finite element model. The mode shape expansion preserves the connectivity of the structure in the final expanded mode shape.
If there is a small change in the stiffness of the structure, each perturbed mode shape can be described as a linear combination of the original mode shapes as
Φd= Φdc
Φdf
=ΦuZ= Φuc
Φuf
Z (4.35)
where the subscript c denotes the master DOFs of the experimental model; the subscriptf denotes the number of additional DOFs to be expanded to form the com-plete mode shape; matrix Z is the transformation matrix; and Φd and Φu are the
mode shape matrices of the damaged and the undamaged structures, respectively.
The cross-orthogonality condition for two mode shapesuanddis written as ΦTuiMΦdj=
m k=1
ΦTuiMΦukZkj=Zij (4.36)
and in a matrix form as
ΦTuMΦd=Z (4.37)
When there is local small damage in the structure,Φuis close toΦd; andZshould be close to a diagonal unity matrix. Therefore, an error function,∆1=I−Z, is used to describe the differences betweenΦuandΦd, whereIis the diagonal unit matrix. This error function includes both the transformation error and the measurement errors.
Another error function describes directly the differences between the measured and the predicted damaged mode shapes from Equation (4.35) as
∆2=Φdc−ΦucZ (4.38)
The two types of error are combined in the form of a weighted Frobenius norm error function expressed as
Ψ = Φdc−ΦucZ2F+WI−Z2F (4.39)
whereW is the weight coefficient. By differentiating this error function with respect to each element of the matrixZand setting it equal to zero, the transformation matrix Zis obtained as
Z=(ΦTucΦuc+WI)−1(ΦTucΦdc+WI) (4.40) Note thatΦuf is not measured and it is practical to use the analytical mode shapes instead. WhenΦuc is not available, i.e. there is no measured information from the undamaged model, the analytical values can also be used in the expansion.
Substituting Equation (4.40) into Equation (4.35) yields the additional DOFs required to form the complete damaged mode shape,
Φdf =ΦufZ (4.41)
This method does not require an accurate analytical model but it should have the correct connectivity assumptions for the structure with appropriate types of finite elements.
The expansion result can be adjusted by weighing the accuracy of the analytical model and the modal analysis model with the weight coefficient,W. If the modal anal-ysis model is reliable and accurate, less weight is given to the analytical information by making the weightWless than 1.0. Otherwise the value ofWis made larger than 1.0.
Note that whenWequals zero, the expansion method converges to the SEREP method 4.6.2 A p p l i c a t i o n
The frame structure (2.82m high and 1.41m wide) is shown in Figure 4.3, with details of the geometrical and physical information and the finite element model. The structure
Detail “A”
7
6
5
4 13 14 15
15 16
17 17
16 18 18
14 12
1 2 3 4 5 6
11 10 12
11 D d
13 b/2
3
2
1 8
9 7 8 10
9
All units in metre B
D b d
Column 0.0688 0.1 0.0638 0.085
Beam 0.0761 0.151 0.0711 0.1359
80 mm
30 mm
Detail ‘A’
40 mm B
E 2.1 1011Nm2, ρ 7800Kgm3
Figure 4.3 Finite element model of two-storey frame structure
is modelled by 18 two-dimensional beam elements of equal length. The beams are connected to the column horizontally by top- and seat-angles and double web-angles with nuts and bolts. Details of this type of semi-rigid beam-column connection are also shown in Figure 4.3. The initial rotational stiffness of the beam-column connection is approximately 3.0×106Nm/rad from static tests. The bottoms of the columns are fully welded to base plates which are welded to the rigid floor, and the same rotational stiffness as for the beam-column connection is assumed for these supports in the finite element modelling.
Table 4.3 MACof the frame structure (a) Damage at node 7
————————————–
Analy. Freq. 1 2 3 4 5 6
Damaged Freq. 22.49 74.17 198.51 221.77 261.32 280.17
1 21.83 0.9979 0.0295 0.0019 0.0023 0.0325 0.0034
2 67.50 0.0053 0.9834 0.0047 0.0019 0.0340 0.0002
3 193.30 0.0004 0.0022 0.9535 0.0002 0.0377 0.0006
4 219.83 0.0001 0.0015 0.0689 0.9173 0.0221 0.0103
5 247.09 0.0145 0.0131 0.0116 0.0598 0.8359 0.1291
6 278.24 0.0000 0.0001 0.0006 0.0017 0.1021 0.7513
(b) Damage at nodes 4 and 7
————————————–
Analy. Freq. 1 2 3 4 5 6
Damaged Freq. 22.49 74.17 198.51 221.77 261.32 280.17
1 18.67 0.9946 0.0373 0.0006 0.0082 0.0206 0.0011
2 67.10 0.0024 0.9796 0.0030 0.0106 0.0032 0.0002
3 191.99 0.0007 0.0032 0.9788 0.0018 0.0115 0.0049
4 217.08 0.0001 0.0001 0.0513 0.7016 0.1336 0.0702
5 235.02 0.0203 0.0153 0.0001 0.2766 0.6690 0.0910
6 273.42 0.0050 0.0017 0.0126 0.0019 0.1182 0.5678
The damage is simulated by removing both the top- and seat-angles at the joint.
This release of restraints only changes the bending stiffness of the horizontal member but retains the original connectivity of the structure. Six modal frequencies and mode shapes of the frame before and after the ‘damage’ is introduced are measured and deter-mined using the Structural Modal Analysis Package. Only the horizontal translational DOFs at nodes 2 to 7 and 9 to 14, and the vertical translational DOFs at nodes 15 to 18 are measured with the intention of collecting information from all parts of the structures. B&K 4370 accelerometers and a B&K 8202 force hammer are used to col-lect the vibration information. The modal frequencies are shown in Table 4.3 and the theoretical mode shapes are shown in Figure 4.4. The incomplete mode shapes are then expanded to the full mode shapes with the modal expansion method described above.
The measured mode shapes are used in the identification, and the analytical mode shapes are used in the mode shape expansion of the incomplete measurement. Any com-parison of the modal data obtained in the two different states requires correct matching of the modes. This is done by calculating the Modal Assurance Criteria (MAC) between the analytical mode shapes and the measured mode shapes. A larger MAC value indi-cates greater correlation in the modes of the two states. The MAC were calculated between the analytical and experimental mode shapes of the frame, and it is larger than or equal to 0.98 for the first two modes in the two damage cases as shown in Table 4.3.
Two damage cases were studied. Case 1 is with damage at node 7. Case 2 is with the same damage at both nodes 7 and 11. The damage location was successfully identified with the first two test mode shapes after the mode shape expansion with 15 analytical modes were included, and the results are shown in Figures 4.5 and 4.6. The damage
f1 22.65Hz f2 74.27Hz f3 22.65Hz
f6 328.54Hz f5 264.80Hz
f4 251.03Hz
Figure 4.4 The theoretical mode shapes of the frame structure
0 0.2 0.4 0.6 0.8 1 1.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
MSECR
Element number Figure 4.5 Identified results for case (a) from two modes
at node 7 was located as damage in element 16 in Figure 4.5, as this beam element is connected to the column at node 7. Damages at nodes 7 and 11 were identified in Figure 4.6 as damages in elements 15 and 16, which are the two beam elements connected to the column at nodes 11 and 7, respectively.
0 0.2 0.4 0.6 0.8 1 1.2
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18
MSECR
Element number Figure 4.6 Identified results for case (b) from two modes