Elishakoff, Isaac; Livshits, David Some closed-form solutions in random vibration of Bernoulli-Euler beams. (English) Zbl 0556.73088 Int. J. Eng. Sci. 22, 1291-1301 (1984). The only closed-form solutions for random vibration of beams are that due to A. Houdijk (1928), for the tip mean-square displacement of a cantilever beam under space- and time-wise ideal white noise, and that due to A. C. Eringen [J. Appl. Mech. 24, 46-52 (1957; Zbl 0081.184)] for a simply-supported beam under identical excitation. In both instances, beams possessing transverse damping were treated. In the present study closed-form solutions are found for uniform, simply supported beams subjected to a stationary excitation that is white both in space and time. The beams possess either structural, Voigt or rotary damping mechanisms. Expressions are obtained for the space-time correlation functions of displacement, velocity and stress. Previously derived interesting conclusions by S. H. Crandall and A. Yildiz [ibid. 29, 267-275 (1962; Zbl 0108.378)] on divergence of the mean-square stress for a beam with Voigt damping, and its convergence for the beam with combined transverse and rotary damping, are confirmed. Moreover, the closed form solution is obtained for the probabilistic characteristics of a beam under a number of separate or combined dampings. Cited in 6 Documents MSC: 74H50 Random vibrations in dynamical problems in solid mechanics 74K10 Rods (beams, columns, shafts, arches, rings, etc.) 74H45 Vibrations in dynamical problems in solid mechanics Keywords:structural damping; closed-form solutions; uniform, simply supported beams; stationary excitation; white both in space and time; rotary damping mechanisms; space-time correlation functions of displacement; velocity; stress; divergence of the mean-square stress; Voigt damping; probabilistic characteristics; separate or combined dampings Citations:Zbl 0081.184; Zbl 0108.378 PDFBibTeX XMLCite \textit{I. Elishakoff} and \textit{D. Livshits}, Int. J. Eng. Sci. 22, 1291--1301 (1984; Zbl 0556.73088) Full Text: DOI References: [1] Houdijk, A., Le Mouvement Brownien d’un Fil, Archives Neerlandaises des Sciences Exactes et Naturelles, Series III A, 11, 212-277 (1928) [2] Eringen, A. C., J. Appl. Mech., 24, 46-52 (1957) [3] Van Lear, G. A.; Uhlenbeck, G. E., Phys. Rev., 38, 1583-1598 (1931) [4] Bogdanoff, J. L.; Goldberg, J. E., J. Aerospace Sci., 371-376 (1960), May [5] Crandall, S. H.; Yildiz, A., J. Appl. Mech., 29, 267-275 (1962) [6] Samuels, J. C.; Eringen, A. C., J. Appl. Mech., 25, 496-500 (1958) [7] Lin, Y. K., Probabilistic Theory of Structural Dynamics (1967), McGraw-Hill: McGraw-Hill New York · Zbl 0359.70052 [8] Elishakoff, I., Probabilistic Methods in the Theory of Structures (1983), Wiley-Interscience: Wiley-Interscience New York · Zbl 0572.73094 [9] Gradshteyn, I. S.; Ryzhik, I. M., Tables of Integrals, Series and Products (1980), Academic Press: Academic Press New York · Zbl 0521.33001 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.