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 and the rev-A waves are associated with (()20)>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)