FE Model Updating for a Turbine Generator Shaft Using Quadratic Taylor Approximations 

Tomas Svensson

Centre for Mathematical Sciences
Mathematical Statistics
Lund Institute of Technology,
Lund University,

The problem posed is to update a model to accurately correspond to some measurements. The model in this particular case is a shaft structure and the measurements are the natural frequencies and the modal shapes. It is critical to have a model that accurately predicts and describes the natural frequencies of the real shaft as running the turbine near these frequencies might damage the shaft. There are several update algorithms for this purpose and the one used here builds on a first order Taylor expansion of the eigenfrequency function. The task of this master thesis is to introduce the second order terms of the Taylor expansion in the model updating algorithm and to compare the results with those of the linear algorithm. To get a good approximation of the measured data one has to iterate in several steps. The calculation of higher order derivatives of the eigenfrequencies is very time expensive and therefore a good update or approximation of the second order derivatives should be found too. The Hessians are approximated by two different approximations formulas called BFGS, after its inventers Broyden Fletcher, Goldfarb and Shanno, and the second is secant SR1 as it is derived from the Secant Relationship. They have in common that they try to reconstruct the Hessians using only function values and first order derivatives. Calculating only a small part, the tri-diagonal, of the Hessian is also investigated. It turns out that the quadratic term increase the stability of the update method and that it also generally update the model with less total change in the parameters that the linear method. It will also be shown that it is enough to analytically calculate an initial Hessian and thereafter update the Hessians with either of the approximating formulas. No significant difference in ``update-performance'' was found between the BFGS and SR1 formulas.