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,
2000:E6

Abstract


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 tridiagonal, 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 ``updateperformance'' was found between the BFGS and SR1 formulas.
