Jtam-A4.dvi JOURNAL OF THEORETICAL AND APPLIED MECHANICS 55, 1, pp. 213-226, Warsaw 2017 DOI: 10.15632/jtam-pl.55.1.213 MODELING AND ANALYSIS OF COUPLED FLEXURAL-TORSIONAL SPINNING BEAMS WITH UNSYMMETRICAL CROSS SECTIONS Jie Wang, Dongxu Li, Jianping Jiang College of Aerospace Science and Engineering, National University of Defense Technology, Changsha, China e-mail: wangjie@nudt.edu.cn The structural modeling and dynamic properties of a spinning beam with an unsymmetri- cal cross section are studied. Due to the eccentricity and spinning, transverse deflections along the two principal directions and the torsional motion about the longitudinal axis are coupled. The structural model of the beam is established based on the Hamilton principle and by incorporating the torsional inertia. Moreover, because of its significant influence on characteristics for the non-circular cross-sectional beam, the warping effect is considered in the formulation. The proposed model is effectively validated in two cases: the spinning be- amwith a symmetric cross section and the cantilevered beamwith an unsymmetrical cross section. Then the effects of the spinning speed on natural frequencies andmode shapes are investigated.Numerical results reveal that the critical speed is alteredwith respect to nonco- incidence of the centroid and the shear center. For the beamswith strongwarping rigidities, the warping effect cannot be neglected due to significant influence on natural frequencies. Keywords: spinning beam, critical speed, warping, coupled flexural-torsional vibration 1. Introduction Spinningbeamsare important components of turbineblades, propellers, elastic linkages, satellite boomsandarewidespread in various branches of structural engineering.Dynamic characteristics such as natural frequencies and mode shapes of these systems are meaningful for analysis of position accuracy, throughput, fatigue and safety. As a consequence, it is essential to accurately establish the dynamic model of spinning beams and predict its vibration characteristics. For the last decades, there has been a growing interest in the investigation of structural modeling, and excellent work has been done on the dynamic analysis of spinning beams. The fullyflexural-torsional couplingmodel for spinningbeamshasbeen successfully establishedbased on the analytical method by Bishop (1959), Dimentberg (1961), Kane (1961), Newland (1972) and Zu and Han (1992). Bishop (1959) utilized Newton’s method to derive the characteristic equation of a bent shaft in the Euler-Bernoulli beam model and investigated the stability of the system. Lagrangian approach (Shiau et al., 2006) andHamilton’s principle (Yoon andKim, 2002) were also utilized to derive the governing equations for the system. Besides, different methods were proposed by researchers in order to solve the governing equations, i.e. assumed- -modesmethod,finite elementmethod, anddynamic stiffnessmethod.Shiau et al. (2006) studied the dynamic behavior of a spinning Timoshenko beamwith general boundary conditions based on the global assumed mode method. Yoon and Kim (2002) utilized the finite element method to analyze the dynamic stability of an unconstrained spinning beam subjected to a pulsating follower force.Banerjee andSu (2004) developed the dynamic stiffnessmethod and theWittrick- -Williams algorithmwas applied to compute natural frequencies andmode shapes. Thismethod was also used in the free vibration analysis of a spinning composite beam (Banerjee and Su, 2006). 214 J.Wang et al. Based on the proposed methods, many researchers dealt with problems of spinning beams subjected to different kinds of loads (Ho and Chen, 2006; Lee, 1995; Zu and Han, 1994) under various boundary conditions (Choi et al., 2000; Zu andMelanson, 1998). Sheu andYang (2005) studied the dynamic response of a spinning Rayleigh beam with rotary inertia and gyroscopic effects in general boundary conditions. The relationship between the critical speed and the hollowness ratio and length-to-radius ratio was investigated by Sheu (2007). Ouyang andWang (2007) presented a dynamic model for vibration of a rotating Timoshenko beam subjected to a three-directional load moving in the axial direction. Popplewell and Chang (1997) investigated free vibrations of a simply supported but stepped spinningTimoshenko beamwith theGalerkin method. Ho and Chen (2006) discussed the vibration problems of a spinning axially loaded pre-twisted Timoshenko beam. Na et al. (2006) established the model of a tapered thin-walled composite spinning beam subjected to an axial compressive force. Moreover, dynamic stability of spinning structures around the longitudinal axis such as a shaft or an unconstrained beam has been widely investigated (Lee, 1996; Tylikowski, 2008). Experimental investigations on a cantilevered spinning shaft have been reported. Qian et al. (2010) conducted a non-contact dynamic testing of a highly flexible spinning vertical shaft. In all of these studies, spinningbeamshad symmetric cross-sections and the shear center and centroidwere assumed to superpose each other. In practical applications, the cross-section of the spinningbeamcanbe eccentric due to errors duringprocessing.Moreover, in some circumstances the cross-section is intended to be eccentric to meet the requirement of the engineering. For a beam with an arbitrary uniform cross section, the coupling of bending and torsion may occur when the beam experiences rotating motion. Yoo and Shin (1998) studied the eigenvalue loci veerings and mode shape variations for a rotating cantilever beam with the coupling effect considered.Latalski et al. (2014) investigated a rotating composite beamwithpiezoelectric active elements.Ananalysis of a rotorwith several flexiblebladeswas conductedwith the spin softening effects and the centrifugal stiffening effects considered through a pre-stressed potential (Lesaffre et al., 2007). Sinha and Turner (2011) further researched the characteristics of a rotating pre- twisted blade. In these studies, the direction of rotation is vertical to the longitudinal direction. Literature focused on the analysis of a beam rotating about its longitudinal direction is few. Filipich et al. (1987) studied the free vibration coupling of bending and torsion of a uniform spinningbeamhaving one axis of symmetry.Then they extended the approach to a beamhaving no symmetric axis anddeveloped adynamicmodel of coupled torsional anbendingdeformations (Filipich and Rosales, 1990). The model accounted for the dynamic coupling terms due to the rotation and the eccentricity. This paper further discusses the bending-torsion coupling effects with the warping effect considered. Natural characteristics including natural frequencies andmode shapes with respect to the spinning velocity and the eccentricity are investigated. And the critical spinning speed variation is observed in the presence of coupling effects. The paper is organized as follows. In Section 2, differential equations of the beam are formulated based on the Hamilton principle. The formulations are built onEuler-Bernoulli beam theorywith thewarping effect and torsional rigidity while neglecting the effect of shear rigidities. In Section 3, we calculate mode shape functions and natural frequencies of the system by applying the assumed mode method. In Section 4, the presentmodel is validated by comparingwith literature andnumerically simulated with examples. The effects of spinning speed and warping on natural frequencies, mode shapes and critical speed are examined. 2. Governing differential equations This Section deals with the formulation of differential equations for a spinning beam with an arbitrary cross section based on Hamilton’s principle. Modeling and analysis of coupled flexural-torsional spinning beams... 215 2.1. System description A homogeneous slender beam with a uniform arbitrary cross-section is depicted in Fig. 1a. The left end of the beam is fixed to abasewhich rotates about the longitudinal axis at a constant angular velocity designated asΩ while the right end is free.When in undeformed configuration, the longitudinal axis of the beam goes through the shear center of the cross section. Fig. 1. Deformed configuration of a spinning beamwith arbitrary cross section Three sets of orthogonal right-handed coordinate frames are defined in order to describe the position vectorR of a differential element dM at a generic pointP. The rectangular coordinate system XYZ is fixed with the inertial frame and the origin O is placed at the shear center of the cross-section on the clamped end. The frame xyz is a rotating framewhose origin o remains coincident with the point O and the x axis remains parallel to the X axis. When the beam spins, directions of the y and z axes are time-varying. The angle between the y axis and the Y axis is represented by the symbolϕ. The third reference frame ξηζ is the element coordinate of the differential beam element which is attached to the shear center S of the beam section. C represents the center of mass of the beam section and (ey,ez) denote the coordinates ofC in the frame Sηζ. The orientations of these frames at a time during free vibration are shown in Fig. 1b. The deformation of a differential beam element located at a distance x from the left end is defined by spatial displacement v(x,t),w(x,t) and rotation φ(x,t) about x-axis. The v(x,t) andw(x,t) represent lateral displacements in the y and z directions, respectively. 2.2. Equations of motion Thegoverning equationof theflexiblebeamis formulatedbasedonthe followingassumptions: (1) for the elementary case of beam flexure and torsion using the Euler-Bernoulli beam theory with torsional inertia but not shear deformation or axial-force effects, (2) the warping effect is considered due to the fact that the torsion inducedwarping occurswhen the beam section is not circular, (3) the axial displacement of the beam is neglected. Moreover, the deformation of the beam is small and yields to the linear conditions. Physical properties of the material are elastic and constant. We consider the spinning beam undergoing transverse displacements and torsional motion. Also, it is assumed that the shear center S and the center of massC of the cross section are not coincident. For such abeam, theposition vector of a representative point after beamdeformation can be defined as R(x) = vj+wk+eyj1+ezk1 (2.1) 216 J.Wang et al. where i, j andk are unit vectors in the x, y and z directions, respectively. And i1, j1 andk1 are unit vectors in the ξ, η and ζ directions, respectively. The velocity of the point can be obtained as follows υ(x)= v̇j+ ẇk+Ωi× (vj+wk)+(Ω+ φ̇)i× (eyj1+ezk1) (2.2) The overhead dot denotes partial derivatives with respect to time t. The kinetic energy can be simplified as T = 1 2 L ∫ 0 ρA(v̇2+ ẇ2) dx+ 1 2 ρJp L ∫ 0 φ̇2 dx+ 1 2 L ∫ 0 ρA[Ω2(v2+w2)−2Ωv̇w+2Ωvẇ] dx + 1 2 L ∫ 0 ρA[(e2y +e 2 z)φ̇ 2−2ezv̇φ̇+2eyẇφ̇+2ezΩwφ̇+2eyΩvφ̇] dx + L ∫ 0 ρA[(e2y +e 2 z)Ωφ̇−ezΩv̇+ezΩ 2w+eyΩẇ+eyΩ 2v] dx + L ∫ 0 ρA(−eyΩv̇φ+eyΩ2wφ−ezΩẇφ−ezΩ2vφ) dx+ 1 2 L ∫ 0 ρA(e2y +e 2 z)Ω 2 dx (2.3) The symbols ρ, E and A denote density, Young’s modulus and cross sectional area. Jp is the polar moment of inertia and is given by Jp = ∫∫ A r2p dηdζ (2.4) where rp represents the distance between a certain point in the section and the center. The potential strain energy of the beam including the warping effect is considered as below U = 1 2 L ∫ 0 E(Izv ′′2+ Iyw ′′2) dx+ 1 2 L ∫ 0 GJpφ ′2 dx+ 1 2 L ∫ 0 EΓφ′′ 2 dx (2.5) where G denotes the shear modulus. Iy and Iz show the second moments of area about the z-axis and y-axis, EΓ is warping rigidity. Primes denote partial derivatives with respect to x. For uniform beams,A, Iy, Iz, Jp andEΓ are constant throughout the span. Then the Lagrangian function of the beam system can be expressed as L=T −U = 1 2 L ∫ 0 ρA(v̇2+ ẇ2) dx+ 1 2 L ∫ 0 ρA[Ω2(v2+w2)−2Ωv̇w+2Ωvẇ] dx + 1 2 ρJp L ∫ 0 φ̇2 dx+ 1 2 L ∫ 0 ρA(e2φ̇2−2ezv̇φ̇+2eyẇφ̇+2ezΩwφ̇+2eyΩvφ̇) dx + L ∫ 0 ρA(e2Ωφ̇−ezΩv̇+ezΩ2w+eyΩẇ+eyΩ2v) dx + L ∫ 0 ρA(−eyΩv̇φ+eyΩ2wφ−ezΩẇφ−ezΩ2vφ) dx+ 1 2 L ∫ 0 ρAe2Ω2 dx − 1 2 L ∫ 0 E(Izv ′′2+ Iyw ′′2) dx− 1 2 L ∫ 0 GJpφ ′2 dx− 1 2 L ∫ 0 EΓφ′′ 2 dx (2.6) Modeling and analysis of coupled flexural-torsional spinning beams... 217 Using Hamilton’s principle, the dynamic model of the system can be obtained EIz ∂4v ∂x4 +ρA(v̈−Ω2v−2Ωẇ−ezφ̈−2Ωeyφ̇+ezΩ2φ)= ρAexΩ2 EIy ∂4w ∂x4 +ρA(ẅ−Ω2w+2Ωv̇+eyφ̈−2Ωezφ̇−eyΩ2φ)= ρAezΩ2 EΓ ∂4φ ∂x4 −GJp ∂2φ ∂x2 +(ρAe2+ρJp)φ̈+ρAey(2Ωv̇+ ẅ−Ω2w) +ρAez(−v̈+Ω2v+2Ωẇ)= 0 (2.7) When skipping the eccentricity of the cross section, Eq. (2.7) has the following form EIz ∂4v ∂x4 +ρA(v̈−Ω2v−2Ωẇ)= 0 EIy ∂4w ∂x4 +ρA(ẅ−Ω2w+2Ωv̇)= 0 EΓ ∂4φ ∂x4 −GJp ∂2φ ∂x2 +ρJpφ̈=0 (2.8) The first two equations in Eq. (2.8) are fully consistent with the results by Banerjee and Su (2004). Also, it can be concluded that the eccentricity induces the coupling between transverse deformations and torsional motion. When skipping the spinning, Eq. (2.7) has the following form, which is consistent with the results by Tanaka and Bercin (1999) EIz ∂4v ∂x4 +ρA(v̈−ezφ̈)= 0 EIy ∂4w ∂x4 +ρA(ẅ+eyφ̈)= 0 EΓ ∂4φ ∂x4 −GJp ∂2φ ∂x2 +(ρAe2+ρJp)φ̈+ρA(−ezv̈+eyẅ)= 0 (2.9) It is obvious that the coupling between v andw takes place due to spinning. 3. Mode shape and frequency equation For a free homogeneous vibration problem, a sinusoidal oscillation is assumed v(x,t) =V (x)ejωt w(x,t)=W(x)ejωt φ(x,t) =Φ(x)ejωt j = √ −1 (3.1) where ω is the circular frequency of oscillation, V , W and Φ are amplitudes of v, w and φ, respectively. Substituting Eq. (3.1) into differential equation (2.7) leads to EIz ρA V (4)− (ω2+Ω2)V −2jωΩW +ez(ω2+Ω2)Φ−2jωΩeyΦ=0 EIy ρA W(4)− (ω2+Ω2)W +2jωΩV −ey(ω2+Ω2)Φ−2jωΩezΦ=0 EΓ ρA Φ(4)− GJp ρA Φ′′− ( e2+ Jp A ) ω2Φ+(ω2+Ω2)(ezV −eyW)+2jωΩ(eyV +ezW)= 0 (3.2) For convenience, we consider a beamwith amonosymmetric cross-sectionwith the symmetry axis y. The centroidC is on the axis y and the scalar ez is equal to zero. Then Eq. (3.2) can be simplified as 218 J.Wang et al. EIz ρA V (4)− (ω2+Ω2)V −2jωΩW −2jωΩeyΦ=0 EIy ρA W(4)− (ω2+Ω2)W +2jωΩV −ey(ω2+Ω2)Φ=0 EΓ ρA Φ(4)− GJp ρA Φ′′− ( e2y + Jp A ) ω2Φ+2jωΩeyV −ey(ω2+Ω2)W =0 (3.3) Then introducing the differential operator D and subsequent variables as follows D= d dx L11 = EIz ρA D4− (ω2+Ω2) L12 =−2jωΩ L13 =−2jωΩey L21 =2jωΩ L22 = EIy ρA D4− (ω2+Ω2) L23 =−ey(ω2+Ω2) L31 =2jωΩey L32 =−ey(ω2+Ω2) L33 = EΓ ρA D4− GJp ρA D2− ( e2y + Jp A ) ω2 (3.4) It can be seen that Y ,Z and Ψ satisfy the equation ∆    V W Φ    =0 (3.5) where ∆=    L11 L12 L13 L21 L22 L23 L31 L32 L33    (3.6) Introducing κ1 = ρA EIz κ2 = ρA EIy κ3 = ρA EΓ κ4 = GJp EΓ κ5 = ( e2y + Jp A ) κ6 =ω 2+Ω2 (3.7) and setting the determinant of differential operator matrix (3.6) equal to zero leads to the following twelvth order differential equation: (D4−κ6κ1)[(D4−κ6κ2)(D4−κ4D2−κ3κ5ω2)−e2yκ2κ3κ 2 6] −4ω2Ω2κ1κ2[(D4−κ4D2−κ3κ5ω2)+e2yκ3κ6]+4ω 2Ω2e2yκ1κ3D 4 =0 (3.8) The solution to the above equation can be expressed in an exponential form R(x)= erx (3.9) Specifying s = r2, then substituting Eq. (3.9) into (3.8), the following characteristic equation can be obtained s6−κ4s5− (κ3κ5ω2+κ2κ6+κ1κ6)s4+(κ2κ4κ6+κ1κ4κ6)s3 +(κ1κ2κ 2 6−4κ1κ2ω 2Ω2−κ2κ3κ26e 2 y −4κ1κ3e 2 yω 2Ω2+κ1κ3κ5κ6ω 2 +κ2κ3κ5κ6ω 2)s2+κ1κ2κ4(4ω 2Ω2−κ26)s−κ1κ2κ3(κ5κ 2 6ω 2−κ36e 2 y +4κ6e 2 yω 2Ω2−4κ5ω4Ω2)= 0 (3.10) Modeling and analysis of coupled flexural-torsional spinning beams... 219 s1-s6 are solutions to Eq. (3.10). The twelve roots of Eq. (3.8) can be written as ±ri ri = j √ si i=1,2, . . . ,6 (3.11) Then the general solutions of V ,W and Φ are expressed as V (x)=A1coshr1x+A2 sinhr1x+A3coshr2x+A4 sinhr2x+A5coshr3x+A6 sinhr3x +A7cosr4x+A8 sinr4x+A9cosr5x+A10 sinr5x+A11cosr6x+A12 sinr6x W(x)=B1coshr1x+B2 sinhr1x+B3coshr2x+B4 sinhr2x+B5coshr3x+B6 sinhr3x +B7cosr4x+B8 sinr4x+B9cosr5x+B10 sinr5x+B11cosr6x+B12 sinr6x Φ(x)=C1coshr1x+C2 sinhr1x+C3coshr2x+C4 sinhr2x+C5coshr3x+C6 sinhr3x +C7cosr4x+C8 sinr4x+C9cosr5x+C10 sinr5x+C11cosr6x+C12 sinr6x (3.12) whereAi,Bi andCi (i=1-12) are three different sets of constants. Substituting Eq. (3.12) into Eq. (3.2), relations between Ai,Bi andCi can be derived B1 = p1A1 B2 = p1A2 B3 = p2A3 B4 = p2A4 B5 = p3A5 B6 = p3A6 B7 = p4A7 B8 = p4A8 B9 = p5A9 B10 = p5A10 B11 = p6A11 B12 = p6A12 C1 = q1A1 C2 = q1A2 C3 = q2A3 C4 = q2A4 C5 = q3A5 C6 = q3A6 C7 = q4A7 C8 = q4A8 C9 = q5A9 C10 = q5A10 C11 = q6A11 C12 = q6A12 (3.13) where pi = κ2κ6 2jωr4iΩ ( 1 κ1 r4i −κ6+ 4ω2Ω2 κ6 ) i=1,2, . . . ,6 qi =                          κ2κ 2 6e( 1 κ1 r4i −κ6+ 4ω2Ω2 κ6 ) +4eyω 2Ω2r4i 2jωr4iΩ ( 1 κ3 r4i − κ4 κ3 r2i −κ5ω2 ) i=1,2,3 κ2κ 2 6e ( 1 κ1 r4i −κ6+ 4ω2Ω2 κ6 ) +4eyω 2Ω2r4i 2jωr4iΩ ( 1 κ3 r4i + κ4 κ3 r2i −κ5ω2 ) i=4,5,6 (3.14) The constantsA1-A12 can be determined from the boundary conditions. For a clamped-free beam, the boundaries are as follows clamped end (x=0) : V =0,V ′ =0,W =0,W ′ =0,Φ=0,Φ′ =0 free end (x=L) : V ′′ =0,V ′′′ =0,W ′′ =0,W ′′′ =0,κ4Φ ′−Φ′′′ =0,Φ′′ =0 (3.15) Using boundary condition (3.15), a set of twelve homogeneous equations in terms of the constants A1-A12 will be generated. The natural frequencies ω can be numerically solved by setting the determinant of the coefficient matrix of A1-A12 to be equal to zero. 4. Numerical applications and results In this Section, firstly some limiting cases are examined to validate the model presented he- re. Secondly, the dynamic characteristics of the beam with unsymmetrical cross sections are investigated using the proposed method. 220 J.Wang et al. 4.1. Validation The example for validating is taken from literature (Banerjee and Su, 2004). The beam has a rectangular cross section and possesses equal flexural rigidities in the two principal directions of the cross section. The properties are given by: EIyy = 582.996Nm 2, EIzz = 582.996Nm 2, ρA=2.87kg/m,L=1.29m. The non-dimensional natural frequency and the spinning speed parameter are defined as in literature (Banerjee and Su, 2004) ω∗i = ωi ω0 Ω∗ = Ω ω0 (4.1) where ω0 = √ √ EIyyEIzz ρAL4 (4.2) Comparison of the first three natural frequencies in the current study with those given in published literature is listed in Table 1. Both examples apply to cantilever end conditions, and the effect of warping stiffness is excluded in the analysis. It is concluded that the resulting frequencies are in good agreement with the one given in the previouswork. BecauseEIyy equals EIzz in this example, the natural frequency parameters of the first two modes are equal when the spinning speed parameter is zero. Table 1. Natural frequencies of the spinning beam: (1) Banerjee and Su (2004), (2) present method Spinning Natural frequency parameters (ω∗i ) speed ω∗1 ω ∗ 2 ω ∗ 3 parameter (Ω∗) (1) (2) (1) (2) (1) (2) 0 3.516 3.516 3.516 3.516 22.034 22.034 2 1.516 1.516 5.516 5.516 24.034 24.034 3.5 0 0 7.016 7.016 25.534 25.534 4 – – 7.516 7.516 26.034 26.034 Then, to investigate characteristics of the beamwith unsymmetrical cross section, two uni- form beams with a semi-circular open cross section and with a channel cross section showed in Fig. 2, are considered. Physical properties of the beams for validation are derived from (Bercin and Tanaka, 1997), as shown in Table 2. Fig. 2. The cross sections of the two beams studied The first seven natural frequencies for the beams given in Fig. 2 are obtained by including and excluding the effect of warping stiffness when the spinning speed is zero, and compared with the results by Bercin and Tanaka (1997), as shown in Table 3. It is observed that when the effect of warping is neglected, the errors associated with it become increasingly large as the modal index increases. Modeling and analysis of coupled flexural-torsional spinning beams... 221 Table 2.Physical properties of the beams studied Parameters Example I Example II EIy [Nm 2] 6380 1.436 ·105 EIz [Nm 2] 2702 2.367 ·105 GJ [N] 43.46 346.71 EΓ [Nm4] 0.10473 536.51 ρ [kg/m3] 2712 2712 A [m2] 3.08 ·10−4 1.57 ·10−3 L [m] 0.82 2.7 e [m] 0.0155 0.0735 Table 3. Natural frequencies [Hz] of the beam: (1) Bercin and Tanaka (1997); (2) present approach including warping; (3) present approach excluding warping Modal Example I Example II index (1) (2) (3) (1) (2) (3) 1 63.79 63.79 62.65 11.03 11.02 8.332 2 137.7 137.7 130.4 – 18.10 18.10 3 – 149.7 149.7 39.02 39.02 23.92 4 278.4 278.4 261.5 58.19 58.20 36.74 5 484.8 484.8 422.5 – 113.4 47.42 6 663.8 663.8 613.3 152.4 152.4 67.41 7 – 768.4 656.3 209.4 209.4 86.64 4.2. Spinning speed To examine the effect of the spinning speed on natural frequencies of the beam with an unsymmetrical cross section, various values with the interval [0,4] for the spinning speed pa- rameter are considered for Example I and Example II, and the corresponding frequencies are presented in Tables 4 and 5. Table 4.Natural frequencies of Example I versus the spinning speed parameter Spinning speed Natural frequency parameters (ω∗i ) parameter (Ω∗) ω∗1 ω ∗ 2 ω ∗ 3 ω ∗ 4 ω ∗ 5 ω ∗ 6 0 2.149 4.639 5.044 9.378 16.333 22.365 1 1.783 4.607 5.453 9.355 16.319 22.340 2 0.760 4.729 6.235 9.287 16.278 22.268 2.25 0 4.783 6.453 9.262 16.264 22.244 3 – 4.988 7.132 9.174 16.212 22.163 4 – 5.327 8.069 9.027 16.122 22.043 It is found that the spinning speed alters the natural frequencies, especially at the lower vibration modes. With an increase of the spinning speed, the coupling between y-axial and z-axial deformations becomes larger,which is demonstrated inEq. (3.3). Therefore,mode shapes of the systemchangedue to larger couplingandnatural frequencies varycorrespondingly.Mostly, as the modal index rises, the effect of spinning speed on natural frequencies weakens since the motion amplitudes become smaller with an increasing frequency, which corresponds to an insignificant change in the reference kinetic energy. 222 J.Wang et al. Table 5.Natural frequencies of Example II versus the spinning speed parameter Spinning speed Natural frequency parameters (ω∗i ) parameter (Ω∗) ω∗1 ω ∗ 2 ω ∗ 3 ω ∗ 4 ω ∗ 5 ω ∗ 6 0 2.426 3.984 8.587 12.809 24.966 33.541 1 1.943 4.461 8.737 12.741 25.038 33.517 2 1.024 5.335 9.166 12.547 25.249 33.448 2.63 0 5.937 9.553 12.372 25.449 33.380 3 – 6.300 9.811 12.255 25.588 33.332 4 – 7.303 10.532 11.971 26.038 33.171 Figures 3a and 3b show variations of the first four non-dimensional natural frequencies with respect to the spinning speed parameter. Because of the large difference between the bending rigidities in the two principal planes, the natural frequencies start off with different values. The fundamental frequencies of both examples decrease with the increasing spinning speedwhile the others decrease or increase. At a certain spinning speed, which is defined as the critical speed, the first natural frequency becomes negative, resulting in instability. For the spinning beam with circular or rectangular cross-section, the natural frequencies are obtained by subtracting or adding the natural frequencies whenΩ∗ =0 to the spinning speed parameter (Banerjee and Su, 2004). So the value of the critical spinning speedwhen the beambecomes unstable equals to the first frequency of the beam with Ω∗ =0. For the spinning beam with an unsymmetrical cross- section, the noncoincidence of mass center and shear center induces coupled flexural-torsional modes and alters the critical speed. Both values of the critical speed are larger than the first frequencies for the examples studied. Fig. 3. Natural frequencies versus the spinning speed for (a) Example I, (b) Example II 4.3. Warping effect The relative errors of natural frequencies due to the warping effect are discussed in this Section. Figures 4a and 4b show changes of natural frequencies with respect to the spinning speed with inclusion and exclusion of the warping for Example I and II, respectively. It is evident that the inclusion of the warping effect increases the natural frequencies. And when the warping effect is neglected, the errors associated with it become increasingly larger as themodal index increases. Additionally, errors inExample II aremore severe than inExample I. This is because the proportion of warping rigidity to bending rigidity in Example II is larger than that inExample I. It is also observed that the exclusion ofwarpingmakes the critical speed decrease. Modeling and analysis of coupled flexural-torsional spinning beams... 223 Fig. 4. Natural frequencies versus the spinning speed for (a) Example I, (b) Example II Fig. 5. Mode shapes in Example I with the speed parameter forΩ∗ =0 4.4. Mode shapes Thefirst four normalizedmodal shape functions inExample I are illustrated inFigs. 5 and 6. It is concluded that in any case, the transverse deflection along the z-axis and torsional motion about thex-axis are coupled.Thefirst twomodes are coupled vibrationmodes of z-axial bending and x-axial torsion, while the thirdmode is the y-axial bendingmode.When the spinning speed parameter is set to 2.0, all themodes become strongly coupled.Moreover, the speed has caused significant changes to the relative amplitudes between the z-axial displacement and x-axial 224 J.Wang et al. torsional angle, especially for lower modes. The beams with unsymmetrical cross-sections show different characteristics compared with symmetric cross-sectional beams, for which the effects of spinning speed onmode shapes are marginal, as declared by Banerjee and Su (2004). Fig. 6. Mode shapes in Example I with the speed parameter forΩ∗ =2 5. Conclusions This paper presents dynamic analysis of a spinning beamwith an unsymmetrical cross section. The governing equations are formulated based on the Euler-Bernoulli beam theory and the Hamilton principle, and then natural frequencies and mode shapes are derived by the assumed modemethod.Effects of the spinning speed andwarping on natural frequencies are investigated. Numerical simulations are conducted in order to validate the present method and some main conclusions are derived as follows. The noncoincidence of centroid and shear center of the unsymmetrical cross section induces the coupling of transverse deflection and torsional motion. The spinning speed induces coupling between transverse deflections along two orthogonal axes. The value of the spinning speed is critical to natural frequencies of the system. The mode shapes are notably changed due to the spinning speed which is different compared to the beamwith the symmetric cross section. Also the critical speed increases for the spinning beamwith unsymmetrical cross sections. It has also shown that the warping effect has a significant influence on the natural frequencies. Moreover, the effects of warping on the natural frequencies become increasingly large when the proportion of warping rigidity to bending rigidity is notable. Modeling and analysis of coupled flexural-torsional spinning beams... 225 References 1. Banerjee J., Su H., 2004, Development of a dynamic stiffness matrix for free vibration analysis of spinning beams,Computers and Structures, 82, 23, 2189-2197 2. 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