PROCEEDINGS -OF- Proc.R.Soc.A(2005)461,3913-3934 THE ROYAL doi:10.1098/rspa.2005.1524 SOCIETY Published online 4 October 2005 Helical structure of the waves propagating in a spinning Timoshenko beam BY K.T.CHAN,N.G.STEPHEN2 AND S.R.REID3 Department of Mechanical Engineering,The Hong Kong Polytechnic University,Hung Hom,Kowloon,HKSAR,People's Republic of China (mmktchan@polyu.edu.hk) 2Mechanical Engineering,School of Engineering Sciences,The University of Southampton,Highfield,Southampton SO17 1BJ,UK 3School of Mechanical,Aerospace and Civil Engineering,The University of Manchester,PO Bor 88,Sackville Street,Manchester M60 1QD,UK The aim of the paper is to study the cause of a frequency-splitting phenomenon that occurs in a spinning Timoshenko beam.The associated changes in the structure of the progressive waves are investigated to shed light on the relationship between the wave motion in a spinning beam and the whirling of a shaft.The main result is that travelling bending waves in a beam spinning about its central axis have the topological structure of a revolving helix traced by the centroidal axis with right-handed or left-handed chirality.Each beam element behaves like a gyroscopic disc in precession being rotated at the wave frequency with anticlockwise or clockwise helicity.The gyroscopic effect is identified as the cause of the frequency splitting and is shown to induce a coupling between two interacting travelling waves lying in mutually orthogonal planes.Two revolving waves travelling in the same direction in space appear,one at a higher and one at a lower frequency compared with the pre-split frequency value.With reference to a given spinning speed,taken as clockwise,the higher one revolves clockwise and the lower one has anticlockwise helicity,each wave being represented by a characteristic four-component vector wavefunction. Two factors are identified as important,the shear-deformation factor g and the gyroscopic-coupling phase factor 0.The q-factor is related to the wavenumber and the geometric shape of the helical wave.The 0-factor is related to the wave helicity and has two values,+/2 and-m/2 corresponding to the anticlockwise and clockwise helicity, respectively.The frequency-splitting phenomenon is addressed by analogy with other physical phenomena such as the Jeffcott whirling shaft and the property of the local energy equality of a travelling wave.The relationship between Euler's formula and the present result relating to the helical properties of the waves is also explored. Keywords:helical wave;propagating wave;four-component vector wavefunction; gyroscopic effect;Timoshenko beam;Euler's formula 1.Introduction A normal mode of vibration of a spinning beam will be described as the superposition of a revolving sa-standing and su-standing wave (Chan et al.2002) Received 28 September 2004 Accepted 8 June 2005 3913 2005 The Royal Society
Helical structure of the waves propagating in a spinning Timoshenko beam BY K. T. CHAN1 , N. G. STEPHEN2 AND S. R. REID3 1 Department of Mechanical Engineering, The Hong Kong Polytechnic University, Hung Hom, Kowloon, HKSAR, People’s Republic of China (mmktchan@polyu.edu.hk) 2 Mechanical Engineering, School of Engineering Sciences, The University of Southampton, Highfield, Southampton SO17 1BJ, UK 3 School of Mechanical, Aerospace and Civil Engineering, The University of Manchester, PO Box 88, Sackville Street, Manchester M60 1QD, UK The aim of the paper is to study the cause of a frequency-splitting phenomenon that occurs in a spinning Timoshenko beam. The associated changes in the structure of the progressive waves are investigated to shed light on the relationship between the wave motion in a spinning beam and the whirling of a shaft. The main result is that travelling bending waves in a beam spinning about its central axis have the topological structure of a revolving helix traced by the centroidal axis with right-handed or left-handed chirality. Each beam element behaves like a gyroscopic disc in precession being rotated at the wave frequency with anticlockwise or clockwise helicity. The gyroscopic effect is identified as the cause of the frequency splitting and is shown to induce a coupling between two interacting travelling waves lying in mutually orthogonal planes. Two revolving waves travelling in the same direction in space appear, one at a higher and one at a lower frequency compared with the pre-split frequency value. With reference to a given spinning speed, taken as clockwise, the higher one revolves clockwise and the lower one has anticlockwise helicity, each wave being represented by a characteristic four-component vector wavefunction. Two factors are identified as important, the shear-deformation factor q and the gyroscopic-coupling phase factor q. The q-factor is related to the wavenumber and the geometric shape of the helical wave. The q-factor is related to the wave helicity and has two values, Cp/2 and Kp/2 corresponding to the anticlockwise and clockwise helicity, respectively. The frequency-splitting phenomenon is addressed by analogy with other physical phenomena such as the Jeffcott whirling shaft and the property of the local energy equality of a travelling wave. The relationship between Euler’s formula and the present result relating to the helical properties of the waves is also explored. Keywords: helical wave; propagating wave; four-component vector wavefunction; gyroscopic effect; Timoshenko beam; Euler’s formula 1. Introduction A normal mode of vibration of a spinning beam will be described as the superposition of a revolving sa-standing and sb-standing wave (Chan et al. 2002) Proc. R. Soc. A (2005) 461, 3913–3934 doi:10.1098/rspa.2005.1524 Published online 4 October 2005 Received 28 September 2004 Accepted 8 June 2005 3913 q 2005 The Royal Society
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)
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 crosssectional 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 lefthanded (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 crosssection 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)
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)
Wave helical structure 3917 11 -ig eil,)for 21 Figure 2.Splitting of a plane-polarized traveling sinusoidal FIS wave (solid line)into a rev-A RH-helix FIS wave (dash line)and a rev-C LH-helix FIS wave (dash-dot line)arising from the frequency-splitting induced by the gyroscopic effect.The rev-A wave revolves at and rev-C wave revolves at w2,where2>.The plane-polarized wave is not revolving before splitting.The curves in this figure are traced by the centroidal axis. where the +and-'signs are for the backward-in-space (BIS)waves and forward-in-space(FIS),respectively.Upper case symbols Wr W,Φ，andΦ， represent the corresponding amplitudes,and k are the angular frequency and wavenumber of the bending wave motion along the beam axis,respectively,@ being positive definite. For a travelling wave with frequency w,the wavenumber k equals k or ky the subscript being used to indicate the direction of the transverse displacement of the wave in the z-or y-axis,respectively.However,k,and ky are equal when the beam is circular.In equation (2.1),ei is taken as a factor to define an interacting relation between the two waves in the mutually orthogonal planes.It will be shown that such interaction manifests as a coupling arising from the gyroscopic effect. Equation(2.1)represents either an anticlockwise wave or a clockwise wave, depending on the sign of 0.Given either value of 6,the bending wave assumes the shape of a perfect helix,traced by the centroidal axis of the beam, revolving at and the orbit of each of the centroidal points is circular.One may refer to e=ti as the helicity of a revolving wave.Thus,an anticlockwise wave has helicity e2=+i and a clockwise wave has helicity e-i/2=-i.Given the helicity,the sign before the wavenumber k in e)determines the handedness of the centroidal helix,left-handed or right-handed corkscrew-like chirality. Proc.R.Soc.A (2005)
where the ‘C’ and ‘K’ signs are for the backward-in-space (BIS) waves and forward-in-space (FIS), respectively. Upper case symbols Wx, Wy, Fx and Fy represent the corresponding amplitudes, u and k are the angular frequency and wavenumber of the bending wave motion along the beam axis, respectively, u being positive definite. For a travelling wave with frequency u, the wavenumber k equals kx or ky the subscript being used to indicate the direction of the transverse displacement of the wave in the x - or y-axis, respectively. However, kx and ky are equal when the beam is circular. In equation (2.1), eiq is taken as a factor to define an interacting relation between the two waves in the mutually orthogonal planes. It will be shown that such interaction manifests as a coupling arising from the gyroscopic effect. Equation (2.1) represents either an anticlockwise wave or a clockwise wave, depending on the sign of q. Given either value of q, the bending wave assumes the shape of a perfect helix, traced by the centroidal axis of the beam, revolving at u and the orbit of each of the centroidal points is circular. One may refer to eiq ZGi as the helicity of a revolving wave. Thus, an anticlockwise wave has helicity eip/2ZCi and a clockwise wave has helicity eKip/2ZKi. Given the helicity, the sign before the wavenumber k in ei(ut G kzCq) determines the handedness of the centroidal helix, left-handed or right-handed corkscrew-like chirality. Figure 2. Splitting of a plane-polarized traveling sinusoidal FIS wave (solid line) into a rev-A RH-helix FIS wave (dash line) and a rev-C LH-helix FIS wave (dash-dot line) arising from the frequency-splitting induced by the gyroscopic effect. The rev-A wave revolves at u1 and rev-C wave revolves at u2, where u2Ou1. The plane-polarized wave is not revolving before splitting. The curves in this figure are traced by the centroidal axis. Wave helical structure 3917 Proc. R. Soc. A (2005)
3918 K.T.Chan and others (c)Equation of motion and four-component vector wave solution (i)Beam properties,coordinate system and equations While the main results will pertain to a circular cross-section beam,to start from an asymmetric cross-section first helps to bring out the 0-factor that can be identified as the value for characterizing a beam.This transpires to be a simple factor in this case,but is otherwise time dependent. The nomenclatures of the beam properties are given first.The beam is assumed to be infinitely long.For an arbitrary cross-section,with respect to the fixed coordinate system in which the z-axis is along the beam axis,the product and second moments of areas about the z-and y-axes,ILand I are not invariant with respect to time. Likewise neither are the shear coefficients K and K with respect to the centroidal translations in x-and y-axes,respectively.The cross-sectional area,the density,the shear modulus of rigidity,and Young's modulus,A,p,G and E,are taken as constant for the uniform and homogeneous beam.The spinning speed is taken as positive according to the conventional right-hand screw rule. Short-form notations for partial differential operators are used,for example, 2=(02/t2),2=(02/022),=(0/at),.=(0/0z),etc.The operator D for the bending wave propagating in the spinning Timoshenko beam is given as pAo?-KGA02 KGAd. 0 0 7 -KGAo. plyo-EI,02 0 2p(L,+I,)d: +KGA D= -pLw明 (2.2) 0 0 pAd:-KyGAd2 KyGAd. 0 -Qp(I:+Iu)o -KyGAd pL07-EL02 -pLy明 +KyGA The equation of motion with respect to D operating on the s of equation(2.1)is written as D8=0, (2.3) where 0={0000}T for free motion of the waves in the beam.Equation (2.3)with operator D written as in equation (2.2)is for a general case of a uniform beam with asymmetric cross-section.It will become apparent that,when the cross-section of the uniform beam is circular,operator D is reduced to a skew-symmetric matrix so that the wavefunctions in equation(2.3)represents a travelling wave with a perfect helix shape traced by the centroidal axis revolving either clockwise or anticlockwise. First,define the amplitude ratios Φ， 9x= W王 and qy= (2.4) By substitution,equation (2.1)becomes W 士iqr Wr ei(wt±kz) ei Wy (2.5) 中x Proc.R.Soc.A (2005)
(c ) Equation of motion and four-component vector wave solution (i) Beam properties, coordinate system and equations While the main results will pertain to a circular cross-section beam, to start from an asymmetric cross-section first helps to bring out the q-factor that can be identified as the value for characterizing a beam. This transpires to be a simple factor in this case, but is otherwise time dependent. The nomenclatures of the beam properties are given first. The beam is assumed to be infinitely long. For an arbitrary cross-section, with respect to the fixed coordinate system in which the z -axis is along the beam axis, the product and second moments of areas about the x - and y-axes, Ixy,Ix and Iy, are not invariant with respect to time. Likewise neither are the shear coefficients kx and ky with respect to the centroidal translations in x - and y-axes, respectively. The cross-sectional area, the density, the shear modulus of rigidity, and Young’s modulus, A, r, G and E, are taken as constant for the uniform and homogeneous beam. The spinning speed U is taken as positive according to the conventional right-hand screw rule. Short-form notations for partial differential operators are used, for example, v2 t Zðv2 =vt 2 Þ; v2 z Zðv2 =vz2 Þ, vtZ(v/vt), vzZ(v/vz), etc. The operator D for the bending wave propagating in the spinning Timoshenko beam is given as D Z rAv2 t KkxGAv2 z kxGAvz 0 0 KkxGAvz rIyv2 t KEIyv2 z 0 UrðIx CIyÞvt CkxGA KrIxyv2 t 0 0 rAv2 t KkyGAv2 z kyGAvz 0 KUrðIx CIyÞvt KkyGAvz rIxv2 t KEIxv2 z KrIxyv2 t CkyGA 2 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 5 : ð2:2Þ The equation of motion with respect to D operating on the s of equation (2.1) is written as Ds Z 0; ð2:3Þ where 0Zf g 0000 T for free motion of the waves in the beam. Equation (2.3) with operator D written as in equation (2.2) is for a general case of a uniform beam with asymmetric cross-section. It will become apparent that, when the cross-section of the uniform beam is circular, operator D is reduced to a skew-symmetric matrix so that the wavefunctions in equation (2.3) represents a travelling wave with a perfect helix shape traced by the centroidal axis revolving either clockwise or anticlockwise. First, define the amplitude ratios qx Z Fy Wx and qy Z Fx Wy : ð2:4Þ By substitution, equation (2.1) becomes s Z wx fy wy fx 8 >>>>>>>: 9 >>>>= >>>>; Z Wx GiqxWx eiq Wy Gieiq qyWy 8 >>>>>>>>>: 9 >>>>>= >>>>>; eiðutGkzÞ : ð2:5Þ 3918 K. T. Chan and others Proc. R. Soc. A (2005)
Wave helical structure 3919 Substituting equation (2.5)into (2.3),differentiating the wavefunction with respect to z and t,and separating the real and imaginary parts,one can write four characteristic equations as (I+)qy2 cos 0+plry g2sin=0, (2.6) p(I Iv)qQoW:[KyGAky-qy(EI ky KyGA-pI2)]Wy sin 0 =0, (2.7) [p(I+I)q20 sin 0-pI cos Wy (2.8) [K GAk:-qr(EI ki+KGA-pIy2)]W:=0, pIryq2W:-[Ky GAky-qu(EIky +Ky GA-pI2)]Wy COs 0 =0. (2.9) For the spinning beam with arbitrary cross-section,it is apparent that II and I change with Ot (or just t when 2 is constant). (ii)The 0-factor for the helicity of waves For a circular beam,the orientation of the cross-section relative to the inertia coordinate system is no longer distinguishable.Thus,I=I=I and I=0.The value zero for Ir leads to the simplification of(2.6)to 2IpgQw cos 0 =0. (2.10) This implies =+(/2)as the only two possible roots for a non-zero 2.Equation (2.9)gives the same result.It has already been mentioned but will be shown later that the two-valued 6,characterizing the helicity of the waves,arises from the gyroscopic effect,and is thus referred to as the gyroscopic-phase 0-factor. Table 1 gives the summary of the helicity of the waves.In the table,the sign of (()/0)is indicative of whether a bending wave in the beam has clockwise (rev-C)or anticlockwise (rev-A)helicity.The frequency w is taken as positive. The speed Q is positive also taking clockwise spinning of beam as the reference. It will be shown later that neither the sign of the exponent of e or e-i,nor the sign of the 0-factor alone can be used to indicate the helicity of the progressive wave.There is a pattern of signs of (()/0),as summarized in table 1,that indicates the helicity of the waves in the beam.It can be seen that, the rev-C waves are associated with (()/)0. (iii)Helicity and chirality of the travelling waves Geometrically,due to a wave passing through a beam section,the centre of the cross-section has a translated position w=Wrer Wyeyr (2.11) where e,and e are unit vectors in the fixed x-and y-axes.For a symmetrical beam,from equations (2.7)and (2.8),it can be shown that Wr=W=Wo.Thus, expression (2.5)for the four-component wave function is simplified to ±iq 8 = W。 ei(wt吐) (2.12) ±ieq Proc.R.Soc.A (2005)
Substituting equation (2.5) into (2.3), differentiating the wavefunction with respect to z and t, and separating the real and imaginary parts, one can write four characteristic equations as rðIx CIyÞqyUu cos q CrIxyqyu 2 sin q Z 0; ð2:6Þ rðIx CIyÞqxUuWx C½kyGAkyKqyðEIx k2 y CkyGAKrIxu 2 ÞWy sin q Z0; ð2:7Þ ½rðIx CIyÞqyUu sin qKrIxyqyu 2 cos qWy C½kxGAkxKqx ðEIyk2 x CkxGAKrIyu 2 ÞWx Z 0; ð2:8Þ rIxyqxu 2 WxK½kyGAkyKqyðEIx k2 y CkyGAKrIxu 2 ÞWy cos q Z 0: ð2:9Þ For the spinning beam with arbitrary cross-section, it is apparent that Ix, Iy and Ixy change with Ut (or just t when U is constant). (ii) The q-factor for the helicity of waves For a circular beam, the orientation of the cross-section relative to the inertia coordinate system is no longer distinguishable. Thus, IxZIyZI and IxyZ0. The value zero for Ixy leads to the simplification of (2.6) to 2IrqUu cos q Z0: ð2:10Þ This implies qZG(p/2) as the only two possible roots for a non-zero U. Equation (2.9) gives the same result. It has already been mentioned but will be shown later that the two-valued q, characterizing the helicity of the waves, arises from the gyroscopic effect, and is thus referred to as the gyroscopic-phase q-factor. Table 1 gives the summary of the helicity of the waves. In the table, the sign of ((Gu)U/q) is indicative of whether a bending wave in the beam has clockwise (rev-C) or anticlockwise (rev-A) helicity. The frequency u is taken as positive. The speed U is positive also taking clockwise spinning of beam as the reference. It will be shown later that neither the sign of the exponent of ei(utGkz) or e Ki(utGkz) , nor the sign of the q-factor alone can be used to indicate the helicity of the progressive wave. There is a pattern of signs of ((Gu)U/q), as summarized in table 1, that indicates the helicity of the waves in the beam. It can be seen that, the rev-C waves are associated with ((Gu)U/q)!0 and the rev-A waves are associated with ((Gu)U/q)O0. (iii) Helicity and chirality of the travelling waves Geometrically, due to a wave passing through a beam section, the centre of the cross-section has a translated position w Zwxex Cwyey; ð2:11Þ where ex and ey are unit vectors in the fixed x - and y-axes. For a symmetrical beam, from equations (2.7) and (2.8), it can be shown that WxZWyZWo. Thus, expression (2.5) for the four-component wave function is simplified to s Z wx fy wy fx 8 >>>>>: 9 >>>= >>>; Z Wo 1 Giq eiq Gieiq q 8 >>>>>>>: 9 >>>>= >>>>; e iðutGkzÞ ; ð2:12Þ Wave helical structure 3919 Proc. R. Soc. A (2005)
3920 K.T.Chan and others Table 1.Clockwise (rev-C)and anticlockwise (rev-A)waves-summary of the helicity 2>0 wavefunction spin 0=x/2 0=-x/2 e(a土(+) rev-Asig(+d)2/0)=(+) rev-C sign ((+)/0)=(-) e-iau生a(-)） rev-C sign ((-W)2/0)=(-) rev-A sign ((-@)2/0)=(+) where q=k-(po2/KGk)as defined by Huang (1961).Given k and w,g is fixed, and once the amplitude of the centroidal displacement of the cross-section from the equilibrium position is known,the amplitude of the angular orientation of the cross-section is fixed.The centroidal displacement is written as w=W。e∠p, (2.13) where Wo=w?+w is the length of the centroidal displacement vector relative to a'point of the cross-section intersecting the fixed z-axis.The displacement direction is denoted by the angle o that can be regarded as the angle of polarization of the wave.Thus, arctan- (2.14) r The bending angle is φ=φrer+中geg=qWe∠po: (2.15) For the FIS wave,substituting expressions in (2.1)or (2.12)into (2.14)and applying Euler's formula yield Wiei(ot-k)io o arctan Wei(wt-tz习 [cos(ωt-kgz+)+isin(ωt-kyz+)] arctan (2.16) [cos(ot-k:2)+isin(@t-k:2)] The ratio of the sine imaginary parts or that of the cosine real parts can be used to obtain the following results.However,choosing to use the real parts for a circular beam,equation (2.16)becomes cos(@t-kz)cos 0-sin(wt-kz)sine o arctan (2.17) cos(wt-kz) For0= sin(wt-kz） 2 arctan cos(wt-kz) (2.18) For 0=7 p=arctan sin(wt-kz) (2.19) cos(wt-kz) Thus,when 0>0,p=-wt+k2, (2.20) Proc.R.Soc.A(2005)
where qZkK(ru2 /kGk) as defined by Huang (1961). Given k and u, q is fixed, and once the amplitude of the centroidal displacement of the cross-section from the equilibrium position is known, the amplitude of the angular orientation of the cross-section is fixed. The centroidal displacement is written as w Z Woe:4; ð2:13Þ where WoZ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi w2 x Cw2 y q is the length of the centroidal displacement vector relative to a point of the cross-section intersecting the fixed z -axis. The displacement direction is denoted by the angle f that can be regarded as the angle of polarization of the wave. Thus, 4 Zarctan wy wx : ð2:14Þ The bending angle is f Zfxex Cfyey ZqWoe:4f: ð2:15Þ For the FIS wave, substituting expressions in (2.1) or (2.12) into (2.14) and applying Euler’s formula yield 4 Zarctan WyeiðutKkyzÞ eiq Wx eiðutKkx zÞ Zarctan ½cosðutKkyz CqÞ Ci sinðutKkyz CqÞ ½cosðutKkx zÞ Ci sinðutKkx zÞ : ð2:16Þ The ratio of the sine imaginary parts or that of the cosine real parts can be used to obtain the following results. However, choosing to use the real parts for a circular beam, equation (2.16) becomes 4 Zarctan cosðutKkzÞcos qKsinðutKkzÞsinq cosðutKkzÞ : ð2:17Þ For q Z p 2 ; 4 Zarctan KsinðutKkzÞ cosðutKkzÞ : ð2:18Þ For q ZKp 2 ; 4 Zarctan sinðutKkzÞ cosðutKkzÞ : ð2:19Þ Thus, when Uu q O0; 4 ZKut Ckz; ð2:20Þ Table 1. Clockwise (rev-C) and anticlockwise (rev-A) waves—summary of the helicity wavefunction spin UO0 qZp/2 qZKp/2 e i(utGkz) (Cu) rev-A sign ((Cu)U/q)Z(C) rev-C sign ((Cu)U/q)Z(K) eKi(utGkz) (Ku) rev-C sign ((Ku)U/q)Z(K) rev-A sign ((Ku)U/q)Z(C) 3920 K. T. Chan and others Proc. R. Soc. A (2005)
Wave helical structure 3921 and when 2w 0,9=-wt-, 2w (2.23) and when 2u<0,p=wt+k2. (2.24) The anticlockwise wave of(2.23)is a LH-helix while the clockwise wave of(2.24) is a RH-helix.It is remembered that both topological structures correspond to the FIS wave. (iv)The gyroscopic precession of the spinning beam elements as the waves travel As shown previously,Wr=W=Wo.Normalizing we take Wo=1.Each point of the helix revolves at the wave angular frequency @tracing a circular orbit of unit length.At each point of the centroidal axis,the cross-section has a bending angular rotation o=geo given by (2.15),being the angle of inclination of the spin axis of each cross-sectional element of the beam with respect to an axis parallel to the z-axis,as shown in figure 1a or b.Thus,the revolving motion of the travelling wave manifests as a precession of every beam element at the wave frequency @The direction of precession depends on the sign of (@/0), anticlockwise when it is positive or clockwise when it is negative as shown in table 1. (d)Frequency splitting due to gyroscopic effect It is noted from equation (2.3)that the system operator D represented by (2.2)is skew-symmetric.The 4X4 matrix is partitioned into four 2X2 sub- matrix blocks.Extracting the two off-diagonal blocks for a circular beam we have 0 0 0 0 (2.25) 0 Proc.R.Soc.A(2005)
and when Uu q !0; 4 ZutKkz: ð2:21Þ Equation (2.20) represents that the displacement vector w has its polarization revolving in the anticlockwise direction, while (2.21) shows that the polarization revolving in the clockwise direction. At any instant t, the anticlockwise wave of (2.20) has the tip of the vector w tracing a right-handed helix (RH-helix) in the z -axis, 4ZKutCkz. Similarly, the clockwise wave of (2.21) has the tip of w tracing a LH-helix, 4ZutKkz. For the BIS wave, swei(utCkz) one may replace equation (2.17) by 4 Zarctan cosðut CkzÞcos qKsinðut CkzÞsin q cosðut CkzÞ : ð2:22Þ Thus, when Uu q O0; 4 ZKutKkz; ð2:23Þ and when Uu q !0; 4 Zut Ckz: ð2:24Þ The anticlockwise wave of (2.23) is a LH-helix while the clockwise wave of (2.24) is a RH-helix. It is remembered that both topological structures correspond to the FIS wave. (iv) The gyroscopic precession of the spinning beam elements as the waves travel As shown previously, WxZWyZWo. Normalizing we take WoZ1. Each point of the helix revolves at the wave angular frequency u, tracing a circular orbit of unit length. At each point of the centroidal axis, the cross-section has a bending angular rotation fZqe:4f given by (2.15), being the angle of inclination of the spin axis of each cross-sectional element of the beam with respect to an axis parallel to the z -axis, as shown in figure 1a or b. Thus, the revolving motion of the travelling wave manifests as a precession of every beam element at the wave frequency u. The direction of precession depends on the sign of (uU/q), anticlockwise when it is positive or clockwise when it is negative as shown in table 1. (d ) Frequency splitting due to gyroscopic effect It is noted from equation (2.3) that the system operator D represented by (2.2) is skew-symmetric. The 4!4 matrix is partitioned into four 2!2 submatrix blocks. Extracting the two off-diagonal blocks for a circular beam we have 0 0 0 2UrIvt " # and 0 0 0 K2UrI vt " #; ð2:25Þ Wave helical structure 3921 Proc. R. Soc. A (2005)
3922 K.T.Chan and others showing the difference in the minus sign.After D operates on the four- component wavefunction of (2.12),we have a new matrix for D written as -pA02+KGAk2 ±iKGAk 0 0 干iKGAk -pIo2+EIk2 0 -2iQwpI +KGA (2.26) 0 0 -pAo2+kGAk2 ±iKGA 0 2iQwpI 干iK GAk -pIo2+EIk2 +KGA Equation (2.3)may then be divided into two.The upper blocks give -pA02+KGAk2 ±ik GAk 干iK GAk -pI02+EIk2+KGA-2iQwpleti(/2) }-8 (2.27 and the lower blocks yield -pA@2+KGAk2 ±ik GAk 干iK GAk w+E]{生}-日} (2.28) The two equations are exactly similar,and thus the frequency equation is simply written as -pAo2+KGAk2 士ik GAk =0 (2.29) 干iK GAk -pIw2+EI2+KGA士22wpl The sign of the diagonal terms (tikGAk)is related to the sign of the FIS or BIS wave expression.However,this does not matter for calculating the dispersion characteristics,thus -pAo2+KGAk2 iK GAk 0 (2.30) -ikGAk -pI02+EIk2 +KGA+2QwpI When the spinning speed is zero,2=0,this equation corresponds to the dispersion relation (2.6)derived by Chan et al.(2002)who predicted dispersion of two types of travelling waves admissible for the non-spinning Timoshenko beam,the sa-wave and the sp-wave.The two types of wave solutions can be considered individually.In this context,however,we do not need to do so.Their propagative characteristics will be affected by the gyroscopic effect in the same way depending on the term +2Qwpl.The Proc.R.Soc.A (2005)
showing the difference in the minus sign. After D operates on the fourcomponent wavefunction of (2.12), we have a new matrix for D written as KrAu2 þ kGAk2 GikGAk 0 0 HikGAk KrIu2 þ EIk2 0 K2iUurI þkGA 0 0 KrAu2 þ kGAk2 GikGAk 0 2iUurI HikGAk KrIu2 þ EIk2 þkGA 2 6 6 6 6 6 6 6 6 6 6 4 3 7 7 7 7 7 7 7 7 7 7 5 ð2:26Þ Equation (2.3) may then be divided into two. The upper blocks give KrAu2 CkGAk2 GikGAk HikGAk KrIu2 CEIk2 CkGAK2iUurI e Giðp=2Þ " # 1 Giq ( ) Z 0 0 ( ); ð2:27Þ and the lower blocks yield KrAu2 CkGAk2 GikGAk HikGAk KrIu2 CEIk2 CkGAC2iUurI e Hiðp=2Þ " # 1 Giq ( ) Z 0 0 ( ): ð2:28Þ The two equations are exactly similar, and thus the frequency equation is simply written as KrAu2 CkGAk2 GikGAk HikGAk KrIu2 CEIk2 CkGAG2UurI Z0: ð2:29Þ The sign of the diagonal terms (GikGAk) is related to the sign of the FIS or BIS wave expression. However, this does not matter for calculating the dispersion characteristics, thus KrAu2 CkGAk2 ikGAk KikGAk KrIu2 CEIk2 CkGAG2UurI Z0: ð2:30Þ When the spinning speed is zero, UZ0, this equation corresponds to the dispersion relation (2.6) derived by Chan et al. (2002) who predicted dispersion of two types of travelling waves admissible for the non-spinning Timoshenko beam, the sa-wave and the sb-wave. The two types of wave solutions can be considered individually. In this context, however, we do not need to do so. Their propagative characteristics will be affected by the gyroscopic effect in the same way depending on the term G2UurI. The 3922 K. T. Chan and others Proc. R. Soc. A (2005)
