University of Wales Swansea, UK

Aston University, UK

The modal model represents a further reduction in the number of data points. But is the quantity and quality of information reduced? The FRF may be reconstructed quite accurately using the modal model, and this is done to check the accuracy of the curve fitting. Some researchers believe that the FRF data contains some information from the out of range modes. While theoretically this might be true, in practice, unless the mode is just out of the range, the modes outside the range are easily corrupted by noise and the response is dominated by the in band modes. Thus the FRF and the modal model essentially contain the same information. The advantage of the modal model is that it may be checked to ensure that the data looks reasonable. The advantage of the FRF is that the data is closer to its raw state and the curve fitting may eliminate effects that would help in the identification of damage. For algorithms based on a predominantly linear model, this is unlikely to be an advantage. The equivalence of the FRF and modal data means that it should be possible to draw the same conclusions when presented with equivalent data. Problems arise mainly because the weights required for the data and the initial FE model to ensure that the identification is equivalent are difficult to determine.

It is commonly acknowledged that natural frequencies may be measured more accurately than mode shapes. Typical resolution for the natural frequencies of a lightly damped mechanical structure are 0.1% whereas typical mode shape errors are 10% or more. The situation is further complicated because mode shapes are relatively insensitive to damage. For example, if a cantilever beam contains a crack, the first bending mode will look very much like the first mode of the undamaged beam, until the damage is very severe. Thus natural frequencies should be weighted very heavily compared to mode shapes in any identification exercise. Mode shapes are still valuable to pair the analytical and experimental modes, and also for 'symmetrical' structures which often have areas which produce similar changes to the natural frequencies.

The major problem for damage location, and indeed for error location in model updating, is that all elements in the matrices may be changed. If only a small number of sites are modelled incorrectly (or are damaged) then only a small number of the matrix elements will be changed. Generally, because of the minimum norm optimisation in the updating method, all the matrix elements would be changed a little, rather than a small number of elements changed substantially. Thus the effect of any damage present would be spread over all the degrees of freedom making location difficult. Kaouk and Zimmerman [4] tried to overcome this problem by ensuring that the change in the stiffness matrix was low rank. This does not ensure that the change in stiffness will be local, as the stiffness change could be global but low rank. Also the method, as proposed, requires the rank of the stiffness change to be equal to the number of modes measured. Thus the localisation of damage often becomes worst when more measured modes are included; not a very satisfactory situation.

It should be apparent by now that methods that update whole mass and stiffness matrices have significant disadvantages, which are not outweighed by the major advantage of not requiring parametric models of the damage mechanisms. It is very unlikely that these methods will prove useful in the majority of structural health monitoring situations.

In model updating, the number of parameters may be reduced by only including those parameters that are likely to be in error. Thus if a frame structure is updated, the model of the beams are likely to be accurate but the joints are more difficult to model. It would therefore be sensible to concentrate the uncertain parameters to those associated with the joints. Also, damage might be more likely in the joints. Even so, a large number of potential parameters may be generated, the measurements may still be reproduced and the parameters are unlikely to be identified uniquely. In this situation all the parameters are changed, and regularisation must be applied to generate a unique solution. Regularisation generally applies extra constraints to the parameter estimation problem to ensure a unique solution. Applying the standard Moore-Penrose pseudo inverse is a type of regularisation where the parameter vector with the minimum norm is chosen. The parameter changes may be weighted separately to give a weighted least squares problem, where the penalty function is a weighted sum of squares of the measurement errors and the parameter changes. Such weighting may also be extended to include minimising the difference between equivalent parameters that are nominally equal in different substructures such as joints.

Although using parametric models can reduce the number of parameters considerably, for damage location there will still be a large number of parameters. Most regularisation techniques rely on minimum norm type solutions that will tend to spread the identified damage over a large number of parameters.

In damage location statistical methods and performance measures have been used that work on a similar principle [6]. Only a limited number of sites are assumed to be damaged, and the model updated based on the reduced number of parameters. This process is repeated for all possible combinations of damage site, and possibly even damage mechanism. The results from all the updated models are compared and the one that best matches the measured data is chosen.

The major problem with both subset selection and the statistical type approach, is that many smaller model updating exercises have to be performed. To optimally derive the best set of parameters, or the best damage location, requires the evaluation of many subsets of parameters. With a large number of parameters looking at all subsets of 2 or 3 parameters can become daunting. For example, if we have 100 potential parameters, then there are 100 subsets of 1 parameter, 4950 subsets of 2 parameters and 161,700 subsets of 3 parameters. Thus sub-optimal methods must be used to derive good, but not necessarily the best, subsets of the parameters.

Neural networks are able to treat damage mechanisms implicitly, so that it is not necessary to model the structure in so much detail. The method can also deal with non-linear damage mechanisms easily. Models are still required to provide the training cases for the networks, and this is their major problem. There will always be systematic errors between the model used for training and the actual structure. For success, neural networks require that the essential features in the damaged structure were represented in the training data. The robustness of networks to these errors has not been tested sufficiently. Of course, the other major problem with both genetic algorithms and neural networks is that they require a huge amount of computation for structures of practical complexity, although these methods are well suited to parallel computation.

The structural health assessment of smart structures is one application where the large number of sensors and actuators produces a huge potential benefit, but also a severe difficulty in data interpretation and processing. Algorithms should be tailored to the application and it is unlikely that a single 'best' method will ever emerge. Non-linear methods have significant potential, but the techniques are just starting to be developed. Health assessment using measured ambient excitation is even more difficult, and ways to increase the information content of the measurements would be valuable. For example, strips could be laid across a road bridge so that passing cars excite the structure in a more predictable way. The optimisation of the location of these 'exciters' would make an interesting research topic.

- Doebling, S. W., Farrar, C. R., Prime, M. B. and Shevitz, D. W.
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*Damage Analysis of Bridge Structures using Vibrational Techniques*, Ph.D. Thesis, Aston University, UK, 1992.