JOURNAL OF SOUND AND VIBRATION Journal of Sound and Vibration 291 (2006) 107–131 Digital holography and Karhunen–Loe`ve decomposition for the modal analysis of two-dimensional vibrating structures Umberto Iemma à , Luigi Morino, Matteo Diez Dipartimento di Ingegneria Meccanica ed Industriale, Universita` Roma Tre, Via della Vasca Navale 79, 00146 Rome, Italy Received 15 March 2004; received in revised form 8 March 2005; accepted 31 May 2005 Available online 24 August 2005 Abstract 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–Loe`ve 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 ARTICLE IN PRESS www.elsevier.com/locate/jsvi 0022-460X/$ - see front matter r 2005 Elsevier Ltd. All rights reserved. doi:10.1016/j.jsv.2005.05.029 à Corresponding author. E-mail addresses: u.iemma@uniroma3.it (U. Iemma), l.morino@uniroma3.it (L. Morino), m.diez@uniroma3.it (M. Diez).