Digital holography and Karhunen–Loève decomposition for the modal analysis of two-dimensional vibrating structures

The aim of this paper is to present the basic theory and preliminary applications of a newly developed formulation for the modal analysis of two-dimensional vibrating structures. This is based on the statistical processing of the data extracted from holographic shots of the vibrating object. Specifically, the elastic displacement field is obtained through digital processing of two series of holographic shots (generated by laser beams in quadrature), and then the Karhunen–Loève decomposition (KLD) technique is used to extract, from the data, base functions that are optimal in the sense of maximum content of energy (as understood in signal theory). The coupling of these two well-assessed techniques represents the main novelty of the present work and yields an experimental methodology characterized by several interesting features. First, the use of holographic images as data source provides a non-invasive technique that allows for an accurate analysis of certain phenomena (such as aeroelastic and acoustoelastic problems) for which instrumentation of the experimental models represents a critical issue. Also, it yields simultaneous three-dimensional information on the whole object domain. Moreover, the KLD provides empirical base functions which coincide, in theory, with the fundamental modes of vibration and requires a relatively inexpensive experimental rig to capture high-frequency modes; these in turn are related to the resolution of the digitized holographic shot, and not to the time-sampling rate. In the present work, the optical holographic process is simulated through a dedicated, in-house developed, computer program. The displacement field has been evaluated analytically for simple two-dimensional structures, such as thin homogeneous rectangular plates and membranes. Preliminary numerical results reveal that the KLD base functions obtained with the numerical simulation coincide, within plotting accuracy, with the exact eigenmodes of the structure. In the simulation of the process, attention is paid to the treatment of the measurement noise, always present in real acquisitions. It is shown that the statistical nature of the KLD ensures that the results are not affected by uncorrelated noise with spacially uniform amplitude, even for a very poor signal-to-noise ratio.