Arch Appl Mech (2014) 84:355–374 DOI 10.1007/s00419-013-0804-z ORIGINAL Isaac Elishakoff · Roberta Santoro Random vibration of a point-driven two-span beam on an elastic foundation Received: 2 July 2013 / Accepted: 15 November 2013 / Published online: 10 December 2013 © Springer-Verlag Berlin Heidelberg 2013 Abstract In this study, we investigate the random vibrations of a point-driven two-span beam of length 2 L on a linear elastic foundation. The normal mode approach is utilized. The natural frequencies are falling in two groups: the first group corresponds to the beam of length L that is simply supported at its ends; the second group is associated with the beam of length L that is simply supported at one end and clamped at the other end. The mean-square velocity of the beam is written in terms of auto- and cross-correlations within groups, as well as in terms of cross-correlations between the modes of two groups. In addition, the two-span beam experiences the clustering of frequencies with increased modal density. The effect of cross-correlations between the modal responses corresponding to closely spaced natural frequencies turns out to be extremely significant. Keywords Elastic foundation · Random vibration · Cross-correlation 1 Introduction Wide-band random vibration of point-driven elastic structures has been the object of considerable attention in recent years. Particularly, references [1–9] considered a large number of uniform structures subjected to a point force, with a time history in the form of a stationary wide-band random process. The distribution of the mean-square response velocity turned out to be surprisingly uniform, except for small zones or narrow bands of enhanced response. For example, if the force acts at x = a on a string or a beam of length L , the zones of enhanced response are at x = a and x = L - a. As the bandwidth of the excitation increases without limit, the relative enhancement tends to 3/2 asymptotically. This pattern was demonstrated experimentally and explained by normal mode analysis [1–3] and by the image-sum technique [4]. The cross-correlations between unlike modes could generally be neglected, since there being little modal overlap due to the light damping and ample separation of the natural frequencies; this resulted in symmetrical distribution of the mean-square response velocity with respect to the midpoint of the string or the beam. This symmetry of the response distribution is also preserved in structures subjected to uncorrelated or fully correlated concentrated forces with the relative enhancement depending on the correlation and number of input force process. A structure where this symmetry is not preserved was considered in Ref. [10] which investigated the wide-band random vibration of a uniform circular cylindrical shell excited by a ring load, uniform along the I. Elishakoff Department of Ocean and Mechanical Engineering, Florida Atlantic University, Boca Raton, FL 33431-0991, USA R. Santoro (B ) Dipartimento di Ingegneria Civile, Informatica, Edile, Ambientale e Matematica Applicata, University of Messina, C.da Di Dio, 98166 Messina, Italy E-mail: roberta.santoro@unime.it Tel: +39-90-3977720 Fax: +39-90-3977480