Wave helical structure 3915 shape,the evolution of the topology of the wave in time and space along the beam axis,the force and moment distributions and energy distributions that cause the changes.All these descriptions are directly related to the dispersion and propagation properties of the waves. The following picture of wave phenomena in a spinning beam emerges. A uniform beam with circular cross-section spins at Q rad s about the fixed z-axis that coincides with the undeformed centroidal axis of the beam at rest.A travelling wave will cause the straight centroidal axis of the beam to deform and trace the shape of a helix revolving at rad s about the z-axis. The helix has a constant radius Wo and a pitch.The revolving motion of the helix is regarded as the integrated motion of the precession of each of the cross- sectional elements spinning at about the helically shaped centroidal axis,called the centroidal helix.If the beam is divided into many individual cross-sectional elements,each of them will spin about an axis normal to the cross-section.The angle between the axis normal to the cross-section and the tangent to the centroidal helix at that location is the shear deformation.The shape of the centroidal helix is maintained by the internal stress field due to bending and shear and the inertia associated with the revolving motion of the wave. The revolving motion of the corkscrew-like centroidal helix described earlier will be shown to conform to the energy-transfer process of a progressive wave. This is intimately related to the application of a local property of a travelling wave that the local kinetic and potential energy are equal.Comparing the present energy equality result with those of Elmore Heald (1985)will show some subtle differences as well as similarities. With the present model,the whirling of a shaft can be viewed as the generation of two types of revolving standing waves in superposition,a revolving sa-standing and sp-standing wave.Each type has a right-handed(RH)and a left- handed (LH)helical wave component,both subject to a synchronous excitation, clockwise or anticlockwise.By this model,the studies of Bishop Parkinson (1968)and Morton (1985)for EB beams can be extended to a spinning Timoshenko beam.However,this will not be considered further herein. 2.Theory of helical waves in a Timoshenko beam (a)Various rotational motions with respect to the inertial-coordinate system A fixed right-handed xyz-coordinate system is used with the z-axis coinciding with the centroidal axis of the Timoshenko beam at rest.All the bending waves (waves for short)travel along the z-axis only.The beam spins about the z-axis when there is no wave travelling.However,the centroidal axis is deformed into a helix (centroidal helix)if a wave travels along the z-axis.The helix represents the shape of the travelling wave,revolving about the z-axis.A number of rotational motions of the beam are defined here for purpose of clarity. The rotational motion of a beam element about the axis normal to its cross- section at the centroid is referred to as its spin,and its speed is called its spinning speed.It is clear that the local direction of spin varies along the centroidal helix.The rotation of a wave is referred to as the rotation of the polarization of the wave.It presents as a revolving centroidal helix as shown in figures 1 and 2.The revolving motion of the helix can be described as the Proc.R.Soc.A (2005)shape, the evolution of the topology of the wave in time and space along the beam axis, the force and moment distributions and energy distributions that cause the changes. All these descriptions are directly related to the dispersion and propagation properties of the waves. The following picture of wave phenomena in a spinning beam emerges. A uniform beam with circular cross-section spins at U rad sK1 about the fixed z -axis that coincides with the undeformed centroidal axis of the beam at rest. A travelling wave will cause the straight centroidal axis of the beam to deform and trace the shape of a helix revolving at U rad sK1 about the z -axis. The helix has a constant radius Wo and a pitch. The revolving motion of the helix is regarded as the integrated motion of the precession of each of the cross￾sectional elements spinning at U about the helically shaped centroidal axis, called the centroidal helix. If the beam is divided into many individual cross-sectional elements, each of them will spin about an axis normal to the cross-section. The angle between the axis normal to the cross-section and the tangent to the centroidal helix at that location is the shear deformation. The shape of the centroidal helix is maintained by the internal stress field due to bending and shear and the inertia associated with the revolving motion of the wave. The revolving motion of the corkscrew-like centroidal helix described earlier will be shown to conform to the energy-transfer process of a progressive wave. This is intimately related to the application of a local property of a travelling wave that the local kinetic and potential energy are equal. Comparing the present energy equality result with those of Elmore & Heald (1985) will show some subtle differences as well as similarities. With the present model, the whirling of a shaft can be viewed as the generation of two types of revolving standing waves in superposition, a revolving sa-standing and sb-standing wave. Each type has a right-handed (RH) and a left￾handed (LH) helical wave component, both subject to a synchronous excitation, clockwise or anticlockwise. By this model, the studies of Bishop & Parkinson (1968) and Morton (1985) for EB beams can be extended to a spinning Timoshenko beam. However, this will not be considered further herein. 2. Theory of helical waves in a Timoshenko beam (a ) Various rotational motions with respect to the inertial-coordinate system A fixed right-handed xyz -coordinate system is used with the z -axis coinciding with the centroidal axis of the Timoshenko beam at rest. All the bending waves (waves for short) travel along the z -axis only. The beam spins about the z -axis when there is no wave travelling. However, the centroidal axis is deformed into a helix (centroidal helix) if a wave travels along the z -axis. The helix represents the shape of the travelling wave, revolving about the z -axis. A number of rotational motions of the beam are defined here for purpose of clarity. The rotational motion of a beam element about the axis normal to its cross￾section at the centroid is referred to as its spin, and its speed U is called its spinning speed. It is clear that the local direction of spin varies along the centroidal helix. The rotation of a wave is referred to as the rotation of the polarization of the wave. It presents as a revolving centroidal helix as shown in figures 1 and 2. The revolving motion of the helix can be described as the Wave helical structure 3915 Proc. R. Soc. A (2005)