The stability of uniform rotation of a rigid body about a principal axis of inertia is analyzed for the case where there is a diametral inertia inequality and there is an elastic restoring mechanism with a diametral stiffness inequality which rotates with the body. This model is an idealization for systems such as a two-bladed propeller rotating on a flexible shaft whose stiffness characteristics are not rotationally symmetric. It is found that many such systems possess unstable speed ranges. The instability may be due to either type of asymmetry alone or due to the interaction of the two. Quantitative analytical results are obtained which relate the unstable speed range to the gyroscopic coupling, the inertia inequality, the stiffness inequality, and the relative orientation of the principal axes of inertia with respect to the principal axes of stiffness. Three-dimensional stability surfaces are plotted to give a qualitative overview of the interplay of the various parameters.
On the Stability of Rotation of a Rotor With Rotationally Unsymmetric Inertia and Stiffness Properties
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Crandall, S. H., and Brosens, P. J. (December 1, 1961). "On the Stability of Rotation of a Rotor With Rotationally Unsymmetric Inertia and Stiffness Properties." ASME. J. Appl. Mech. December 1961; 28(4): 567–570. https://doi.org/10.1115/1.3641784
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