Quantification of Vibration Localization in Periodic Structures

Abstract

The phenomenon of vibration mode localization in periodic and near periodic structures has been well documented over the past four decades. In spite of its long history, and presence in a wide range of engineering structures, the approach to detect mode localization remains rather rudimentary in nature. The primary way is via a visual inspection of the mode shapes. For systems with complex geometry, the judgment of mode localization can become subjective as it would depend on visual ability and interpretation of the analyst. This paper suggests a numerical approach using the modal data to quantify mode localization by utilizing the modal assurance criterion (MAC) across all the modes due to changes in some system parameters. The proposed MAC localization factor (MACLF) gives a value between 0 and 1 and therefore gives an explicit value for the degree of mode localization. First-order sensitivity based approaches are proposed to reduce the computational effort. A two-degree-of-freedom system is first used to demonstrate the applicability of the proposed approach. The finite element method (FEM) was used to study two progressively complex systems, namely, a coupled two-cantilever beam system and an idealized turbine blade. Modal data is corrupted by random noise to simulate robustness when applying the MACLF to experimental data to quantify the degree of localization. Extensive numerical results have been given to illustrate the applicability of the proposed approach.