Revolving superposed standing waves in a spinning Timoshenko Beam

The aim of the present paper is to study the revolving superposed standing waves in a spinning Timoshenko beam based on the results obtained by the first author with others in two recently published papers. The concept of superposed standing wave as normal mode and the knowledge about the helical feature of propagating wave will be used to depict a physical picture about the vibration of the spinning beam, from describing the basic constituent waves to showing the orthogonality property of the revolving normal modes. The wave-mechanics approach will be invoked throughout the study with an algebraic procedure used to reveal the two eigenvalues of the gyroscopic-coupling phase factor. A table for tabulating the phases of the centroidal positions of the beam in time and space due to the passing of a wave is invented, through which one could show the helical structure of the wave without ambiguity. From the present result, two types of revolving standing waves are identified, each manifesting as a gyroscopic precession in association with a frequency-splitting phenomenon, in either the clockwise or the anticlockwise direction. It is shown that the revolving waves should be represented by wavefunctions in a form of four-component column matrix vectors.