3914 K.T.Chan and others in the beam.Although the present study is focussed on the travelling wave,a review of the more practical,standing wave structure will be given first,since a standing wave is a combination of two oppositely travelling identical waves. Previous studies (Bishop 1959;Bishop Parkinson 1968;Morton Johnson 1980;Morton 1985)were restricted to using Euler-Bernoulli beam (EB)theory and were mainly concerned with synchronous vibration due to unbalanced masses.More recently,these have been supplemented with studies using Timoshenko beam (TB)theory on vibration,leading to the present study of wave motion in spinning shafts. Using EB theory,Bauer (1980)identified Coriolis inertia effects as the cause of the doubling of the number of natural frequencies of a spinning beam.Lee et al. (1988)used Rayleigh beam (RB)theory and then Zu Han (1992)used TB theory to show that the gyroscopic effect is important. However,their theoretical studies are phenomenological,being focused only on modal descriptions of shaft vibration.Their results did not show the interesting helical properties of the waves that constitute the normal modes they predicted using the standard method for finding modal solutions.The study of wave phenomena in a spinning beam is important because the influence of the gyroscopic effect on the waves underlies the frequency-splitting phenomenon as studied by many previous investigators. Argento Scott (1995),Kang Tan (1998)and Tan Kang(1998)sought flexural wave solutions for a spinning Timoshenko beam.They were able to describe the reflection,transmission and dispersion characteristics of the waves in the spinning beam.However,their results concerning gyroscopic precession are not very clear.One should distinguish between the precession of the spinning beam per se and the precession of a wave propagating in it.From the present study,it is shown that the precession of the wave is free motion that manifests as a corkscrew-like helix traced by the centroidal axis of the spinning beam,which revolves about the axis of the beam at rest to transfer the kinetic and potential energy with the wave,forward or backward along the beam.For clarity, revolving or precession is used for the rotation about the axis of the beam at rest to differentiate it from rotation of a cross-sectional beam element about a transverse axis passing through the centroid in the plane of cross-section.For a helical bending shape,a neutral plane does not exist and so one should be cautious in using the usual definition of a neutral axis,see $2.The above studies also failed to show the helical properties and the gyroscopic-precession motion of bending waves travelling in the spinning beam.This leaves the behaviour and structure of the travelling waves in the spinning beam and their relation to the observed behaviour of the shafts to be explored. In order to address the latter,a new approach is used that is related to the wave-mechanics approach employed by Chan et al.(2002).Using this approach, one may describe the phenomena in terms of their constituent waves,their dispersion behaviour and their relationship to the normal modes.However,the wave-mechanics approach to the spinning beam problem is quite detailed and is worthy of a separate treatment in its own right,which the first author is undertaking.The approach used here does not rely on this new treatment,and is based on the concept of superposed standing waves in a Timoshenko beam (Chan et al.2002).The solution obtained by using this provides a description of the physical phenomena in terms of the geometrical structure of the wave Proc.R.Soc.A (2005)in the beam. Although the present study is focussed on the travelling wave, a review of the more practical, standing wave structure will be given first, since a standing wave is a combination of two oppositely travelling identical waves. Previous studies (Bishop 1959; Bishop & Parkinson 1968; Morton & Johnson 1980; Morton 1985) were restricted to using Euler–Bernoulli beam (EB) theory and were mainly concerned with synchronous vibration due to unbalanced masses. More recently, these have been supplemented with studies using Timoshenko beam (TB) theory on vibration, leading to the present study of wave motion in spinning shafts. Using EB theory, Bauer (1980) identified Coriolis inertia effects as the cause of the doubling of the number of natural frequencies of a spinning beam. Lee et al. (1988) used Rayleigh beam (RB) theory and then Zu & Han (1992) used TB theory to show that the gyroscopic effect is important. However, their theoretical studies are phenomenological, being focused only on modal descriptions of shaft vibration. Their results did not show the interesting helical properties of the waves that constitute the normal modes they predicted using the standard method for finding modal solutions. The study of wave phenomena in a spinning beam is important because the influence of the gyroscopic effect on the waves underlies the frequency-splitting phenomenon as studied by many previous investigators. Argento & Scott (1995), Kang & Tan (1998) and Tan & Kang (1998) sought flexural wave solutions for a spinning Timoshenko beam. They were able to describe the reflection, transmission and dispersion characteristics of the waves in the spinning beam. However, their results concerning gyroscopic precession are not very clear. One should distinguish between the precession of the spinning beam per se and the precession of a wave propagating in it. From the present study, it is shown that the precession of the wave is free motion that manifests as a corkscrew-like helix traced by the centroidal axis of the spinning beam, which revolves about the axis of the beam at rest to transfer the kinetic and potential energy with the wave, forward or backward along the beam. For clarity, revolving or precession is used for the rotation about the axis of the beam at rest to differentiate it from rotation of a cross-sectional beam element about a transverse axis passing through the centroid in the plane of cross-section. For a helical bending shape, a neutral plane does not exist and so one should be cautious in using the usual definition of a neutral axis, see §2. The above studies also failed to show the helical properties and the gyroscopic-precession motion of bending waves travelling in the spinning beam. This leaves the behaviour and structure of the travelling waves in the spinning beam and their relation to the observed behaviour of the shafts to be explored. In order to address the latter, a new approach is used that is related to the wave-mechanics approach employed by Chan et al. (2002). Using this approach, one may describe the phenomena in terms of their constituent waves, their dispersion behaviour and their relationship to the normal modes. However, the wave-mechanics approach to the spinning beam problem is quite detailed and is worthy of a separate treatment in its own right, which the first author is undertaking. The approach used here does not rely on this new treatment, and is based on the concept of superposed standing waves in a Timoshenko beam (Chan et al. 2002). The solution obtained by using this provides a description of the physical phenomena in terms of the geometrical structure of the wave 3914 K. T. Chan and others Proc. R. Soc. A (2005)