Striking Influence of Lumped Masses with Concentrated Rotary Inertia on the Random Vibration Characteristics of an Axially Loaded Beam

The study investigates the random response characteristics of an axially loaded beam carrying lumped masses with inherent rotary inertia. A generalized mathematical model is developed using the Timoshenko-Ehrenfest beam theory in conjunction with the normal mode method to derive the mean square displacement and velocity as a summation of both the direct- and cross-correlation terms. Novel results and dynamic characteristics emerge when the beam system is excited by a point load whose time-history corresponds to band-limited white noise. Namely, it is shown that when a beam carries two or more concentrated masses, modal cross-correlations can contribute significantly to the overall response depending on the magnitude of the inherent rotary inertia and location of the point force, even if the damping is light. Additionally, it is demonstrated that although the loading is broadband, cross-correlations between just two modes is sufficient to significantly increase the drive-point mean square velocity and introduce substantial asymmetry in the spatial distribution of the response. In such cases the response is in stark contrast to the response distribution of point excited uniform beams and strings, which are well-known to be characterized by negligible cross-correlations due to low modal overlap. It is further shown that in cases where the influence of modal cross-correlations is strong, increasing the static tensile load increases the contribution of the cross-terms relative to the direct terms.