The dynamics of a spinning elastic disk with massive load

Abstract

This thesis analyzes the dynamics of a spinning elastic disk. The disk rotates at a constant angular velocity and is acted upon by a load consisting of a mass distributed over a finite area of the disk, a spring, and a dashpot. Using a finite mode approximation, the equation of motion of the transverse deflection of the disk is written as a system of ordinary differential equations with constant coefficients. Analysis of the eigenvalues of the finite mode approximation yields four distinct types of instability. An instability occurs due to the stiffness of the load, terminal instabilities occur due to both the mass of the load and the damping of the load, and an instability occurs as a result of modal coupling. The multiple mode approximation used in the spinning disk analysis is applied to a stationary disk with a moving load. Comparison of the spinning and stationary disk shows the influence of the centrifugal stress of the rotating disk. The direct stability methods of Liapunov are applied to the equation of motion for both the spinning and stationary disk and are used to prove the stability of the systems at speeds below certain critical speeds. Upper bounds for the difference between eigenvalues of the finite mode approximation and eigenvalues of a full infinite mode system are derived. These bounds are calculated for the eigenvalues of a modal coupling instability arising from the finite mode analysis to show that the solution of the full set of equations is also unstable.

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