The percent error in the square root of the forward difference estimate of the derivative, given by exact exact ( ) 100 c c O O - Oc (14) is plotted in Fig.1 vs the % change in the magnitude of the pole for the 6th and 8th modes. As can be seen, the error is relatively small in the 6th modes but significant in the 8th. Since error in the sensitivity approach is directly dependent - one anticipates good performance of the sensitivity scheme in the 6th mode but poor results on the 8th. Figure1. Result from Eq.14 vs change in the magnitude of the pole for two modes of the numerical example Characterizing the Error The relative error in the scaling constant in simulation k is exact exact (k) (k) 100 § · N N K ¨ ¸ ¨ ¸ © N ¹ (15) We examine performance in terms of the mean and standard deviation of K obtained from Monte Carlo simulations. In the simulations the real and the imaginary parts of the identified eigenvalues are assigned lognormal distributions with mean values equal to the true values and prescribed standard deviations. In particular, the coefficient of variation of the real part of the eigenvalue is taken as 0.06, and for the imaginary part we take it as 0.003. Eigenvector variability is included by multiplying each entry of the arbitrarily scaled modes by independently selected vectors whose magnitudes and phases are taken from uniform distributions with limits of {0.95 and 1.05} and {-5 and 5} degrees, respectively. The relative error K is a random variable whose distribution for any mode depends on the magnitude of the mass perturbation. Since the error in the estimate of the derivative does not, however, depend directly on the mass change but on the shift in the pole we use the change in the imaginary part of the pole as the independent variable to display results. Given that K is a random variable either RBN or sensitivity may be most accurate in any given realization. A reasonable approach to contrast performance, however, is to look at what limits of K bound the majority of the realizations. We select the largest of the mean plus or minus two standard deviations which guarantees, accepting the Gaussian premise, that the error is smaller that this limit at least 95% of the time. Fig.2 presents the results for the real and the imaginary parts of the noted limit for the 6th mode and Fig.3 for the 8th. As expected, in the 6th mode the performance of RBN and sensitivity is similar but in the 8th, where the relation between the eigenvalue and the mass perturbation magnitude is highly nonlinear the RBN result is significantly better. 5 0 5 -20 -10 0 10 20 0 5 -20 -10 0 10 20 mode#1 mode#8 mode#6 ( ) - ( ) - ( ) - ( ) - ( ) - ( ) - % change in the magnitude of the pole 396
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