Session: 08-12-01: Optimization, Uncertainty and Probability

Paper Number: 165674

Dramatic Effect of Cross-Correlations on the Random Vibration of an Axially Loaded Beam Carrying Lumped Masses With Rotary Inertia 

A model is presented to determine the response of an axially loaded beam carrying lumped masses with inherent rotary inertia when subjected to a random point load whose time-history corresponds to band-limited white noise. Using the Timoshenko-Ehrenfest beam theory in conjunction with the normal mode method, expressions for the mean square response levels are derived as a summation of both the direct- and cross-correlation terms. Providing that the damping is light and the natural frequencies are well-spaced, the cross-terms may be neglected when computing the statistical response of a randomly excited system. A uniform beam easily meets these conditions since the frequency spacing between successive modes increases monotonically with mode number. Consequently, for a constant damping bandwidth, the modal overlap factor decreases inversely with frequency. In this study, the change in random vibration response when lumped elements are added to an otherwise uniform beam is investigated, with special interest being devoted to the contribution and effects of modal cross-correlations.

The first system considered is that of a simply supported beam of length (L) carrying a central lumped mass with rotary inertia. For this system the modal overlap factor is consistently small regardless of the magnitude of the lumped mass and rotary inertia. When subjected to a broadband random excitation, the corresponding influence of modal cross-correlations is negligible compared to the direct correlations, despite the location of the force. The spatial distribution of the response is symmetric about the center of the beam and two narrow zones of enhanced response levels appear within the vicinity of the point of excitation (x_o) and its image point (L-x_o). Notably, although adding a deterministic axial force to the beam changes the drive-point response significantly, the contribution of the modal cross-correlations remains negligible. Specifically, increasing the axial force in tension is shown to reduce the drive-point response while increasing its magnitude in compression has the opposite effect. Moreover, as the axial force approaches the Euler buckling load, the response increases without bound.

In the second set of systems the beam supports multiple lumped masses with inherent rotary inertia. Here the situation appears quite different from the first case. Namely, it is demonstrated that when a beam supports multiple symmetrically distributed masses, the magnitude of the lumped rotary inertia as well as the location of the point force imposes a considerable influence on the contribution of the cross-correlation component of the response. In some cases, it is shown that modal cross-correlations can contribute as much as 30% to the overall response. In these circumstances it is shown that modal cross-correlations introduce a substantial degree of asymmetry in the spatial distribution of the response while also significantly increasing the drive-point response. Upon including an axial load, it is also observed that in cases where the influence of modal cross-correlations is strong, increasing the tensile load also increases the relative contribution of the cross-terms compared to the direct terms.

Presenting Author: Richard Bachoo University of the West Indies

Presenting Author Biography: Richard Bachoo is currently a Lecturer in the Department of Mechanical and Manufacturing Engineering at The University of the West Indies, St. Augustine. Dr. Bachoo served as a Mechanical and Piping Engineer with the international oil and gas consultancy firm, WorleyParsons for several years before returning to academia in 2017. His research interests include Flow Induced Vibration, High Frequency Structural Vibration and Probabilistic Structural Dynamics.

Authors:

Richard Bachoo University of the West Indies
Isaac Elishakoff Florida Atlantic University