Random Vibration of Beams

Abstract

The calculated response of a uniform beam to stationary random excitation depends greatly on the dynamical model postulated, on the damping mechanism assumed, and on the nature of the random excitation process. To illustrate this, the mean square deflections, slopes, bending moments, and shear forces have been compared for four different dynamical models, with three different damping mechanisms, subjected to a distributed transverse loading process which is uncorrelated spacewise and which is either ideally “white” timewise or band-limited with an upper cut-off frequency. The dynamic models are the Bernoulli-Euler beam, the Timoshenko beam, and two intermediate models, the Rayleigh beam, and a beam which has the shear flexibility of the Timoshenko beam but not the rotatory inertia. The damping mechanisms are transverse viscous damping, rotatory viscous damping, and Voigt viscoelasticity. It is found that many of the mean-square response quantities are finite when the excitation is ideally white (i.e., when the input has infinite mean square); however, some of the responses are unbounded. For these cases the rate of growth of the response as the cut-off frequency of the excitation is increased is obtained.