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Some closed-form solutions in random vibration of Bernoulli-Euler beams. (English) Zbl 0556.73088

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.

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
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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
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