"Two Topics in Nonlinear Mechanics: Critical Speeds and Bouncing Goats"

Lawrie Virgin, Department of Mechanical Engineering and Materials Science, Duke University

The determination of critical speeds of a rotating shaft is a crucial issue in a variety of industrial applications ranging from turbomachinery to disk storage systems. The modeling and analysis of rotor dynamic systems is subject to a number of complications, but perhaps the most important characteristic is to pass through a critical speed under spin-up conditions. This is associated with classical resonance phenomena and high amplitudes, and is often a highly undesirable situation. However, given uncertainties in the modeling of such systems it can be very difficult to estimate based on purely theoretical considerations. Thus, it is clearly useful to gain knowledge of the critical speed of a rotordynamic system under in situ conditions. The first part of this talk will describe a relatively simple approach to predicting the critical speed using data from low rates of revolution. The approach is shown to work well for a variety of rotor dynamics models, and also using experimental data.

The second part of the talk (which is entirely distinct from the first part) considers the deformation and vibration of highly flexible loops in a gravitational field. Both upright and hanging loops are considered. Good agreement is obtained between some theoretical results (based on an inextensible elastica model) and experimental data. An interesting phenomenon of adjacent, co-existing solutions is also described. One of the motivations for this study comes from a fascinating YouTube video.

Professor Virgin's research is centered on studying the behavior of nonlinear dynamical systems. This work may be broadly divided into two components. First, investigation of the fundamental nature of nonlinear systems based on a mathematical description of their underlying equations of motion. Both analytical and numerical techniques are used with special attention focussed on the loss of stability of dynamical systems. The recent discovery of chaos has stimulated much research in this area across the breadth of science and engineering.

The second area of interest is to apply recent results from nonlinear dynamical systems theory to problems of practical engineering importance. These include the nonlinear rolling motion of ships leading to capsize; buckling of axially-loaded structural components; aeroelastic flutter of aircraft panels at high supersonic speeds; vibration isolation based on nonlinear springs; energy harvesting; damage detection and structural health monitoring; and the dynamics of very flexible structures including solar sails and marine risers. Professor Virgin conducts mechanical experiments to complement these studies.

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