PSI - Issue 24

M. Barsanti et al. / Procedia Structural Integrity 24 (2019) 988–996

993

6

M. Barsanti / Structural Integrity Procedia 00 (2019) 000–000

1400

1200

1000

800

600

400

200

0

-2

-1

0

1

2

10 6

Fig. 4. An example of frequency distribution of k xx deviations.

0.63

-2

18

-4

0.405

0.62

16

-6

14

0.61

0.4

-8

12

-10

0.6

0.395

10

-12

0.59

8

-14

0.39

210

140

1

1.2

200

120

190

100

0.95

180

1.15

80

170

60

0.9

160

1.1

40

150

0.85

20

140

1.05

0

130

0.2 0.4 0.6 0.8 1 1.2

0.2 0.4 0.6 0.8 1 1.2

0.2 0.4 0.6 0.8 1 1.2

0.2 0.4 0.6 0.8 1 1.2

Fig. 5. Sti ff ness coe ffi cients as a function of the relative excitation frequency for two di ff erent shaft rotational speeds. The dashed bar indicates systematic error (interval of deviation from the reference value for 95% of the calibration procedures). Note that the scales are di ff erent in each panel to highlight the amplitudes of the systematic error bars with respect to the di ff erent ranges of variation of the sti ff ness coe ffi cients.

6. Comparison of random and systematic errors A detailed analysis of the random error associated to the estimated dynamic coe ffi cients is reported by Barsanti et al. (2019). The simplest method for random error computation is based on the calculation of the stan dard deviation of the sample made of 30 values of the dynamic coe ffi cients. Assuming that the Central Limit Theorem holds (that is to say that 30 is a su ffi ciently high sample numerosity to consider it as infinite), the standard deviation of the estimated dynamic coe ffi cients (averaging the 30 values) is obtained by dividing each sample standard deviation by √ 30. The 95% confidence interval half-width can finally be obtained by multiplying the standard deviation of the mean by 1.96. Barsanti et al. (2019) showed that a more complex methods for the estimate of random error gives substantially the same results. In this section the two types of error are compared reporting the amplitudes of the two sides of the confidence intervals (with respect to each estimated mean value) on the same figure. Both upper and lower extremes are presented, to show that in some cases there is a slight asymmetry for systematic error. This could be due to a slight unbiasing of the method for the systematic error estimation. The results for sti ff ness coe ffi cients are reported