Modeling and dynamic characteristics of spinning Rayleigh beams

This paper derives linear and nonlinear models of spinning Rayleigh beams and investigates dynamic characteristics of downward vertical spinning Rayleigh beams with six different sets of boundary conditions. The derivations show that an important linear term due to centrifugal forces is missing in many reports in the literature because of inconsistent use of nonlinear terms in the derivation, but this term is indispensible for correct modeling and analysis of spinning beams. The influences of rotary inertia, spinning speed, Coriolis and centrifugal forces, slenderness, and gravity on forward and backward whirling speeds, whirling mode shapes, and critical speeds are investigated in detail using analytical and finite-element methods. Numerical results show that each forward whirling speed increases up to its critical speed and the corresponding backward whirling speed decreases when the spinning speed increases. Each forward whirling speed is higher than the corresponding backward whirling speed, and the difference increases when the spinning speed and the mode number increase and the slenderness ratio decreases. For a spinning Rayleigh beam with any of the six different sets of boundary conditions, there are infinite (instead of finite, as some reports said) forward critical speeds and infinite backward critical speeds. Moreover, gravity increases the forward and backward whirling speeds of a downward spinning uniform beam, but the increase is significant only if it is a very long thin beam.