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Estimation of the geometry of randomly dimensioned components from their eigenfrequencies via a multiple linear regression analysis

: Heinrich, M.; Rabe, U.; Hirsekorn, S.; Grabowski, B.

Sas, P. ; Katholieke Universiteit Leuven, Departement Werktuigkunde:
ISMA 2014, International Conference on Noise and Vibration Engineering. Proceedings. CD-ROM : USD 2014, International Conference on Uncertainty in Structural Dynamics; 15 to 17 September, 2014, Leuven, Belgien
Leuven: Katholieke Universiteit Leuven, 2014
ISBN: 978-90-73802-91-9
International Conference on Noise and Vibration Engineering <26, 2014, Leuven>
International Conference on Uncertainty in Structural Dynamics (USD) <5, 2014, Leuven>
Fraunhofer IZFP ()

Acoustic resonance testing (ART) is a non-destructive, integral, and volume-oriented procedure allowing a fast in-line quality assessment of components, e.g. forged or casted metallic parts. ART identifies different characteristic parameters such as eigenfrequencies and damping of the parts by analysing their natural vibration behaviour which mainly depends on geometry and material properties, but also on structural defects, e.g. cracks. These parameters are used as test statistics. Based on an empirical calibration and with the help of a large training set each inspected part can be classified, either as good or as unqualified depending on the values of the test statistics. In addition to the abovementioned modal parameters, many other test quantities can be determined by analysing the natural vibrations of a part, but the eigenfrequencies are the most important. Generally, the exact geometric dimensions and the exact material properties of single parts in a serial production vary randomly within acceptable ranges, for example because of manufacture-related effects, entailing variations in the eigenfrequencies and damping constants of the good parts. These effects are superposed by changes in the parameters caused by defects impeding a reliable classification of the components with the help of ART. A compensation method of those random perturbations is required. This contribution shows how the exact dimensions of defect-free components which are randomly selected out of the same serial production can be estimated from their measured eigenfrequencies. The method is demonstrated by means of 800 virtual connecting rods of constant homogeneous and isotropic material properties. The dimensions of each of the parts with respect to the 8 geometric parameters are described by independent normally distributed random numbers. Using these dimensions, the first 17 eigenfrequencies of each part were calculated numerically by finite element method. A stepwise multiple linear regression analysis was used to estimate the correlations between the geometry parameters and the eigenfrequencies of 600 of the parts. Finally, these correlations and the eigenfrequencies of the remaining 200 parts were used to estimate their exact dimensions. It was found, that the estimated values differ only little from the exact dimensions. We may conclude that the complex effects of limited geometric variations on the eigenfrequencies of a component approximately can be compensated by linear regression functions. Adaptation and implementation of the described procedure to real parts is discussed.