## Finite Element Model Updating in Structural Dynamics

The Finite Element Method is used extensively to model the dynamics of structures and with care is capable of producing accurate results. Often the inaccuracies in the model will arise because of poorly known boundary conditions, unknown material properties or simplification in the modelling. For example, welded or bolted joints will not be rigid but have a finite stiffness which is difficult to predict. Or a plate containing numerous small holes may be approximated by a homogenous plate. These uncertainties in the modelling process cause the predicted dynamics of a structure to be different from the measured dynamics of the real structure. If accurate measured data is available then this data could be used to improve the numerical model in general, and the uncertain parameters of the model in particular. This is model updating.

Recently there has been considerable interest in model updating and a large number of techniques have been proposed. Broadly speaking the methods may be split according to the type of measured data they use and model parameters that are updated. The measured data may be in form of frequency response function (FRF) data or natural frequencies and mode shapes. The updating process may estimate physical parameters, complete mass, damping and stiffness matrices or groups of individual matrix elements. Research so far has concentrated on updating physical parameters using either FRF or modal data. Other aspects of model updating, such as parameter uniqueness, efficient computation, parameterisation, ill-conditioning and the use of incomplete data, are being investigated. The measured data will always be incomplete because the measurements will only be taken at a relatively small number of locations and over a limited frequency range. Issues of parameter uniqueness arise because an infinite number of parameter values can often give rise to identical measured data. The equations to identify the parameters directly will often be ill conditioned and the original finite element model is used to provide extra information and to update the parameters. The process of identification from poorly conditioned equations is called regularization.

One major challenge is to apply the technology to practical problems with a large number of degrees of freedom. Hence the need for computationally efficient programs is growing. Furthermore the key to success in model updating is the choice of parameters. These should be chosen so that the measured data is sensitive to changes in the parameters, but also there should be some indication that there may be modelling errors in the corresponding regions of the model. Good predictability of the updated model depends on choosing a small number of the most important parameters. Modelling joints is particular difficult, and equivalent models for these, often using geometric or generic parameterisations, is a significant application area.

Recently the quantification of parameter uncertainty has become an important research topic. Production techniques require that many structures made to a particular design are nominally identical. However samples taken on real structures demonstrate that there will be a significant variability from structure to structure. As for deterministic model updating, the parameters that model joints cannot be measured directly. Thus if measurements are made on multiple structures, the uncertainty in the parameters may be identified, leading to the concept of stochastic model updating and uncertainty quantification.

The papers listed below give more information.

### Selected References

#### Uncertainty Quantification

Last updated February 2009 by MI Friswell