3916 K.T.Chan and others r/2 3π/2 Wo centreline displaced centroidal axis (a) (b) Figure 1.(a)Distributed gyroscopic elements of the spinning beam induced by a progressive bending wave.(b)Gyroscopic precession of a spinning beam element (enlarged size). gyroscopic precession of each of the spinning beam elements as illustrated in figure 16.The rotation (in radians)of a beam cross-section o about a transverse axis is referred to as bending rotation.Its amplitude is and direction as shown in the same figure.In the figure,O'z'is parallel to the Oz axis. (b)Definitions of helicity and chirality of bending waves The helical properties of the wave will be deduced by solving the equation of motion of a spinning Timoshenko beam.The cross-section of the beam is assumed uniform along the axis but initially the shape is taken as asymmetric. Although the result of the present study can be applied to beam cross-section with rotational symmetry about its centroidal axis,for purposes of brevity,we will use the circular cross-section.When the cross-section is circular,an important factor 6 of e is identified as two-valued and equal to either positive or negative r/2,being associated with the revolving directions of the wave concerned,written as the four-component vector wavefunction W红 W 士i④g 8= , ei(wt土) (2.1) Wy Dr ±iei0Φ Proc.R.Soc.A(2005)gyroscopic precession of each of the spinning beam elements as illustrated in figure 1b. The rotation (in radians) of a beam cross-section f about a transverse axis is referred to as bending rotation. Its amplitude is Fo and direction as shown in the same figure. In the figure, O0 z 0 is parallel to the Oz axis. (b ) Definitions of helicity and chirality of bending waves The helical properties of the wave will be deduced by solving the equation of motion of a spinning Timoshenko beam. The cross-section of the beam is assumed uniform along the axis but initially the shape is taken as asymmetric. Although the result of the present study can be applied to beam cross-section with rotational symmetry about its centroidal axis, for purposes of brevity, we will use the circular cross-section. When the cross-section is circular, an important factor q of eiq is identified as two-valued and equal to either positive or negative p/2, being associated with the revolving directions of the wave concerned, written as the four-component vector wavefunction s Z wx fy wy fx 8 >>>>< >>>>: 9 >>>>= >>>>; Z Wx GiFy eiq Wy Gieiq Fx 8 >>>>>< >>>>>: 9 >>>>>= >>>>>; e iðutGkzÞ ; ð2:1Þ Figure 1. (a) Distributed gyroscopic elements of the spinning beam induced by a progressive bending wave. (b) Gyroscopic precession of a spinning beam element (enlarged size). 3916 K. T. Chan and others Proc. R. Soc. A (2005)