An investigation on the capability of proper orthogonal modes in determining the natural frequencies and damping ratios of linear structural systems

Abstract

The method of Proper Orthogonal Decomposition (POD) is one of the popular techniques used in structural dynamics. Unlike Linear Natural Modes (LNMs), Proper Orthogonal Modes (POMs) derived from the POD method, depend on the excitation loads in addition to the mass and stiffness of the dynamical system. According to the various researches in establishing a general connection between POMs and their respective LNMs, it has been found that except in some special cases, the natural system mode shapes cannot be extracted from the POMs. Despite efforts to find the general relation between POMs and LNMs, the use of POMs in the determination of natural frequencies and damping ratios has received less attention. This paper aims to use the POMs as a modal analysis tool to extract the natural frequencies and damping ratios of linear dynamical systems from their experimental vibrational responses. For this purpose, a POD-based algorithm is proposed to achieve this objective. In the mentioned methodology, at first, the equation of motion of a MDOF system is decomposed into several SDOF equations using proper orthogonal modes. Then the responses of the SDOF models are calculated in proper modal coordinates. Finally, the Power Spectrum Density (PSD) and the Auto Correlation Function (ACF) of their responses are used to determine the natural frequencies and damping ratios, respectively. The accuracy and precision of the results are evaluated by numerical and experimental validation. Two different conventional loading states, including impulse loads and ambient vibrations, are studied. Good agreement was observed between the POD-based results and those obtained from the FE and two other conventional operational modal analysis (OMA) methods. Also, it was found that in the modal identification process by POD-based algorithm, the noise effects are transmitted to the latest proper modes, so in noisy conditions, the accuracy of the modal characteristics corresponding to the primary natural modes will not be affected. Some other findings of this research will be presented at the end of the article.