A nonlinear model of a curved beam for t

Abstract A nonlinear model of curved, prestressed, no-shear, elastic beam, loaded by wind forces, is formulated The beam is assumed to be planar in its reference configuration under its own weight and static wind forces The unit extension, two bending curvatures and the torsional curvature are taken as strains and then expanded up-to second-order terms in the three displacement components and in the angle of twist The incremental equilibrium equations around the prestressed reference configuration are derived, in which shear forces are condensed via a perturbation procedure By using a linear elastic constitutive law and accounting for inertial effects, the complete equations of motion are obtained They are successively strongly simplified by estimating the order of magnitude of all their terms, under the hypotheses of small sagto-span ratio, high slenderness, compact section and by neglecting tangential inertia forces and inertia torsional couples A system of two integro-differential equations in the two transversal displacements only is drawn A simplified model of aerodynamic forces is then developed according to the quasi-static theory The nonlinear, nontrivial equilibrium path of the cable subjected to increasing static wind forces is successively evaluated, and the influence of the angle of twist on the equilibrium is discussed Then stability is studied by discretizing the equations of motion via a Galerkin approach and analyzing the small oscillations around the nontrivial equilibrium Analysis of the limit cycle (galloping) arising at the Hopf bifurcation is left for future investigation Finally, the role of the angle of twist on the dynamic stability of the cable is discussed Keywords: Cables, galloping, twist in cables, aeroelasticity, instability, bifurcation 1 Introduction The analysis of galloping oscillations of iced cables requests a careful formulation both of the mechanical model and of the aeroelastic forces, especially concerning nonlinear regimes (Luongo & Piccardo [1]) The forces are usually modelled referring to the quasi-steady theory, and they depend on the mean wind speed and on the angle of attack, which in its turn descends from the velocity of the structure and from its flow exposition The structure is generally modelled as a perfectly flexible cable, that is as a one-dimensional continuum capable of translational displacements only (Luongo et al [2], Lee & Perkins [3]) This assumption is reliable since the torsion stiffness of the cable is usually very high and the bending stiffness is negligible, respect to the geometric one, because of the slenderness of the structure However, simplified models of cables have highlighted the importance of the twist angle on the determination of aerodynamic forces and, therefore, on the dynamical behavior of the system In particular, Yu et al [4] have introduced the torsion ignoring the initial curvature of the cable, and neglecting all the mechanical nonlinearities Luongo & Piccardo [5] have tried to correct the classic model, adding to the elastic potential energy of the flexible cable an energy of pure torsion ignoring, still, every term of mechanical coupling Therefore, the formulation of a consistent cable-beam model is a matter of great interest, able to take into account all the stiffnesses involved in the problem To the best of authors’ knowledge, similar models are usually employed in fully numerical approaches (for instance, Diana et al [6]) but they are confined to the linear range and are not yet employed in semi-analytical analysis, like the one here proposed A first approach to the subject, devoted to the linear problem, has been presented by the authors in [7] In this paper a nonlinear model of curved elastic prestressed beam, subjected to aerodynamic forces induced by wind, is formulated By taking into account the high slenderness of the body, the model is remarkably simplified via an analysis of the magnitude orders af all the terms in the equations of motion As major result, it is shown that, at the leading order, the dynamic behavior of the cable is governed by the same equations of the perfectly flexible model, in which, however, the positional and velocity-dependent forces also depend on the angle of twist, which is an integral function of the transversal displacements In other words, the twist is a passive variable, slave of the normal and binormal translations The reduced model thus obtained permits to investigate the critical and postcritical aeroelastic behavior of the cable, by highlighting the role of torsion on the stability of the structure Numerical results so far obtained are limited to the bifurcation analysis; the postcritical behavior will be investigated in the next future, along the lines of previous works by the authors [1, 5, 8] The paper is organized as follows The complete equations of motion are derived in Section under the hypotesis of no-shear deformation of the beam In Section a reduced model is drawn from the complete one by neglecting small terms and statically condensing the tangent translation and the angle of twist; therefore, two integrodifferential equations, only in the transversal displacements, are obtained In Section an approximate model for the aerodynamic forces acting on the cable, consistent with the appoximations introduced, is developed In Section the nontrivial equilibrium path of the cable subjected to static wind forces is evaluated, and bifurcations causing galloping (Hopf bifurcations) are detected Numerical results are discussed in Section 6, and some conclusions are drawn in Section Mechanical model Let us consider the cable as a beam constituted by a flexible centerline and rigid crosssections The actual configuration C of the body is described by the curve x = x(s, t) and by the field of basis β := {a1 (s, t), a2 (s, t), a3 (s, t)}, where x is the position occupied by the material point P at the (undeformed) abscissa s at the time t, and β is the inertial principal triad, with a1 normal to the cross section (Fig 1a ) Let us assume as reference configuration C¯ of the body at the time t = the planar ¯ := {¯ ¯ = x ¯ (s) and the field of basis β ¯ (s, t), a ¯ (s, t)} coincident curve x a1 (s, t), a ′ ¯1 ≡ x ¯ the tangent, a ¯ the normal and a ¯ the binormal with the Frenet triad, with a ′ to the curve at s (where ( ) = d/ds) Therefore, according to the Frenet formulas, ¯ ′1 = κ ¯2 , a ¯′2 = −¯ ¯1 , a ¯′3 = 0, where κ a ¯a κa ¯=κ ¯ (s) is the modulus of the curvature vector ¯ ¯ =κ ¯3 in C κ ¯a The referential description of C respect to C¯ requires the assignment of the displacement vector field u(s, t) and of the rotation tensorial field R(s, t), such that: ¯+u, x=x = R¯ , i = 1, 2, (1) where the independent variables s and t have been omitted In particular, the orthog¯ to match the triad β; it admits onal tensor R describes a rotation that leads the triad β ¯ the following scalar representation (both in β and β): [R] = cos ϑ2 cos ϑ3 sin ϑ1 sin ϑ2 cos ϑ3 −cos ϑ1 sin ϑ3 cos ϑ1 sin ϑ2 cos ϑ3 +sin ϑ1 sin ϑ3 cos ϑ2 sin ϑ3 cos ϑ1 cos ϑ3 +sin ϑ1 sin ϑ2 sin ϑ3 cos ϑ1 sin ϑ2 sin ϑ3 −sin ϑ1 cos ϑ3 − sin ϑ2 sin ϑ1 cos ϑ2 cos ϑ1 cos ϑ2 (2) where ϑ3 , ϑ2 and ϑ1 are three elementary rotations ordinately assigned around the (updated) homonymous axes The slenderness of the beam suggests to neglect the shear deformation; therefore the cross-sections are assumed to remain orthogonal to the centerline in any configuration This internal constraint is expressed by the condition x′ = (1 + ε)a1 , a1 being the normal to the section and dx/dS = x′ /(1 + ε) the unit vector tangent to the strained centerline at the actual abscissa S = S(s), with ε := dS/ds − the unit extension By accounting for Eqs (1) and (2) tan ϑ2 = − w′ [(1 + u′ − κ ¯ v)2 + (v ′ + κ ¯ u)2 ] , tan ϑ3 = v′ + κ ¯u ′ 1+u −κ ¯v (3) are derived, together with: ε = (1 + u′ − κ ¯ v)2 + (v ′ + κ ¯ u)2 + w ′2 −1 ≃ u′ − κ ¯v + (v ′ + κ ¯ u)2 + w ′2 (4) ¯ Due to the constraints (3), the configuwhere u, v, w are the components of u in β ration variables u, v, w, ϑ1, ϑ2 , ϑ3 are reduced to the three translation components and the unique rotation component ϑ := ϑ1 , referred as the twist angle In addition to the unit extension, bending and torsion must be introduced as fur¯ of the incremental ther measures of strain They are defined as the components in β curvature vector: ˆ := RT κ − κ ¯ κ (5) ¯ re¯ in the bases β and β, i.e as the difference between the components of κ and κ spectively By using: ¯′i = κ ¯ ×a ¯i a′i = κ × , a (6) ˆ turns out which define the curvature vectors in the two bases, together with Eq (1)b , κ to be the axial vector of the skew-simmetric tensor (see Appendix A): ˆ := RT R′ K (7) ˆ in a two-terms Taylor series and accounting for By expanding the components of κ the constraints conditions (3), it follows: κ ˆ = ϑ′ + κ ¯w′ + κ ¯ vw ′ + w ′ v ′′ + κ ¯ ′ uw ′ ′ κ ˆ = −w ′′ + κ ¯ ϑ + [(u′ − κ ¯ v)w ′] + ϑ [(¯ κu)′ + v ′′ ] ′ κ ˆ = v ′′ + (¯ κu)′ + ϑw ′′ − κ ¯ (ϑ2 + w ′2 ) − [(¯ κu + v ′ )(u′ − κ ¯ v)] (8) The equilibrium equations are then derived By considering an infinitesimal element of cable in the actual configuration (Fig 1b ), and denoting by t(s, t) and m(s, t) the internal contact force and couple, respectively, acting at abscissa s at time t, the balance equations, in Lagrangian form, read: t′ + b = , m′ + x′ × t + c = (9) where b := b(s, t) and c := c(s, t) are body forces and couples densities per undeformed arc-length, including inertial effects It is assumed that in the planar reference ¯ configuration C¯ the cable is loaded by body forces b(s) and no couples: ¯c(s) ≡ By neglecting flexural effects in its own plane, the cable is stressed in C¯ exclusively by ¯ = By subtracting this latter ¯1 and m ¯ = 0, with ¯t′ + b axial forces, namely ¯t = T¯a from Eq (9)a , the incremental equilibrium equations are obtained: ˆ=0, (t′ − ¯t′ ) + b m′ + x′ × t + cˆ = (10) ˆ := b − b ¯ and cˆ := c − c¯ By letting t = (T¯ + Tˆ1 )a1 + Tˆ2 a2 + Tˆ3 a3 , with b ˆ a1 + M ˆ a2 + M ˆ a3 , using Eq (6)a and (1)b to express the derivatives of a′ m=M i ¯ and by recalling that, from Eqs (1)b and (6)b , x′ = a ¯1 + u′ = (1 + u′ − κ in β, ¯ v)¯ a1 + ¯ read: ¯3 , Eq (10), projected onto β, (v ′ + κ ¯ u)¯ a2 + w ′ a T ′ − T¯κ ¯ (v ′ + κ ¯ u) − T2 κ ¯ + b1 + h.o.t = T2′ T3′ M1′ M2′ M3′ ′ + T¯(v ′ + κ ¯ u) + T1 κ ¯ + b2 + h.o.t = ′ ′ ¯ + (T w ) + b3 + h.o.t = − M2 κ ¯ + c1 + h.o.t = + M1 κ ¯ − T3 + c2 + h.o.t = + T2 + c3 + h.o.t = (11) ˆ i and h.o.t stands for “higher order terms” where hats have been omitted on Tˆi and M In Eqs (11), T2 and T3 are reactive shear forces associated with the internal constrains of zero-shear In order to express the equilibrium equations in terms of configuration variables, they must be condensed To this end, Eqs (11)e,f are solved for the unknows T2 and T3 by applying a perturbation method First, all the kinematic quantities are rescaled to introduce a small perturbation parameter ǫ; then T2 and T3 are expanded in ǫ-series up-to ǫ3 -order, and the relevant perturbation equations are solved in chain Finally, the shear forces are replaced in the remaining four equations and only terms up-to ǫ3 -order retained The whole procedure has been performed by the symbolic manipulation software Mathematica [9], and the relevant equations have found to be quite cumbersome; therefore, only their linear part is explicitely given here: T ′ − T¯κ ¯ (v ′ + κ ¯ u) + M ′ κ ¯ + ˜b1 + h.o.t = ′′ −M3 M2′′ M1′ + T¯(v + κ ¯ u) + T1 κ ¯ + ˜b2 + h.o.t = + (M1 κ ¯ )′ + (T¯ w ′ )′ + ˜b3 + h.o.t = ′ ′ (12) − M2 κ ¯ + c1 + h.o.t = where ˜bi are modified forces bi including the effects of c2 and c3 couples B C¯ ϑ A S s a2 ¯2 a P u a3 ¯3 P a ¯ ¯1 a ¯ x t + dt C a1 ¯+u x=x C −m (a) −t m + dm dS b c (b) Figure 1: (a) configurations of the cable; (b) forces and couples on an infinitesimal element A linear, uncoupled, elastic law is then assumed among the stress components in ¯ namely: the actual base β and the strain components in the reference base β, T1 = EAε , M1 = GJκ1 , M2 = EI2 κ2 , M3 = EI3 κ3 (13) where hats have been omitted on κ ˆ i , and EA, GJ, EI2 and EI3 are axial, torsional and flexural stiffnesses of the cable With Eqs (13), the equations of motion (12) read: ′ EA(u′ − κ ¯ v)′ + EI3 κ ¯ [v ′′ + (¯ κu)′ ] − T¯ κ ¯ (v ′ + κ ¯ u) + b1 mă u + h.o.t = ′ ′′ EA¯ κ(u′ − κ ¯ v) − EI3 [v ′′ + (¯ κu)′ ] + T¯(v ′ + u) + b2 mă v + h.o.t = ′ EI2 (−w ′′ + κ ¯ ϑ)′′ + GJ [¯ κ(¯ κw ′ + ϑ′ )] + (Tw ) + b3 mwă + h.o.t = GJ(¯ κw ′ + ϑ′ )′ − EI2 κ ¯ (−w ′′ + κ ¯ ϑ) + c1 − J1 ă + h.o.t = (14) where inertia forces have been made explicit, m being the mass linear density and J1 the inertia polar moment of the section It is worth nothing that Eqs (14) are block uncoupled in their linear part, i.e in-plane small oscillations of the cable are independent of out-of-plane small oscillations, these latter involving torsion However, nonlinearities couple all these equations; moreover, if the forces ˜bi depend on the configuration variables, as it occurs for the aerodynamic forces, even the linear equations are coupled Equations (14) must be sided by suitable boundary conditions If the cable is restrained at both ends by sferical hinges, displacements and moments must vanish there, namely: u = 0, GJ(ϑ′ + κ ¯ w ′ ) + h.o.t = v = 0, EI2 (−w ′′ + κ ¯ ϑ) + h.o.t = (15) ′′ ′ w = 0, EI3 [v + (¯ κu) ] + h.o.t = at s = 0, l The problem is completed by the initial conditions; here it is assumed that the body is at rest at t = Reduced equations of motion The equations of motion previously obtained are too complicated to be treated analytically; therefore, a simplified model is developed in this Section First, the classical hypothesis of small sag-to-span ratio d/l is introduced [2, 3, 10], commonly accepted for cables falling into the technical range Then, advantage is obtained from the fact the cable is a very slender body; hence the flexural-torsional effects are expected to be smaller than the funicular effects, perhaps except close to the boundaries On the other hand, due to the (small but finite) initial curvature, the bending moment contributes to the moment equilibrium around the tangent to the cable, so that bending and torsion couple 3.1 Magnitude order analysis According to the previous ideas, an analysis of the magnitude orders of all the terms in the equilibrium Eqs (14) and boundary conditions (15) is performed along the following lines By assuming that the cable is flat, and forces in the reference configuration are uniformly distributed, the static profile is well described by a parabola Consistenly: d l ≪ 1, T¯ (s) ≃ T¯ = const, κ ¯ (s) ≃ 8d = const l2 (16) By taking into account the slenderness of the cable, and the smallness of T¯ /A in comparison with the elastic modulus E, it follows: EI2,3 =O EAl2 r2 ≪ 1, l2 T¯ ≪ 1, EA O(EI2) = O(EI3 ) = O(GJ), (17) EI2,3 ≪1 T¯l2 where r is a characteristic dimension of the (compact) cross-section Inequality (17)d holds for almost all real cables Then, it needs to estimate the magnitude order of the displacement component ratios It is assumed that: u=O d v , l ϑ=O w , d v = O(w) (18) together with: ∂nu u =O n , n ∂s l and : ∂nv v =O n , n ∂s l ∂ϑ =O ∂s ∂nw w =O n , n ∂s l ∂2ϑ =O ∂s2 d w , l3 d w l4 n = 1, 2, (19) (20) Equation (18)a is suggested by the results of the linear theory of small vibrations [10], and on the fact that u → in the (prevalently) transversal motions (v ≫ u) when d/l → Equations (19) also follow from the linear theory, when the trigonometric nature of the eigenfunctions is recognized Estimates (18)b and (20) are instead drawn by inspection of the solution of the linearized Eq (14)d and the relevant boundary conditions (see Appendix B) Equation (18)c is self-explaining It is worth noting that, due to the different boundary conditions, the translations u, v and w must vanish at the ends, and therefore they are fastly varying functions, whereas the twist angle ϑ, being different from zero at the ends, can vary in a much slower manner As illustrated in detail in Appendix B, ϑ is indeed a slowly varying function in symmetrical modes (in which ϑ = O (w/d)), and again a fastly varying function in antisymmetrical modes (in which ϑ = O (wd/l2 )) The upper estimate of ϑ has been adopted for any motions, in order to account also for nonsymmetrical modes of cables supported at different levels By using previous estimations in Eqs (14) and retaining only dominant terms among linear, quadratic and cubic terms, separately, the following reduced equations are obtained, in the EI2 = EI3 (=: EI) case (circular cross-section): ′ 1 u=0 EA u − κ ¯ v + v ′2 + w + b1 mă 2 1 1 ¯ v + v ′2 + w ′2 )v ′ ]′ + EA κ ¯ (u′ − κ ¯ v + v ′2 + w ′2 ) + [(u′ − κ 2 2 + T¯v ′′ + b2 mă v=0 1 EA (u κ ¯ v + v ′2 + w ′2 )w ′ 2 ′ ′ (21) + T¯ w ′′ + b3 mwă = GJ EI ¯ ϑ + (EI + GJ)¯ κw ′′ − EI κ ¯ ϑv ′′ + GJ(v ′′ w ′ ) + c1 J1 ă = As major result of the analysis, Eqs (21)a,b,c are identical to that of the flexible cable [3], while Eq (21)d represents an additional nonlinear equation in the twist angle If the body forces are independent of ϑ or even zero (as it happens in the free vibrations), the translational motion is independent of ϑ, which is therefore a passive variable, slave of the translations In contrast, if the body forces depend on ϑ, as in the aerodynamic case, the twist angle does affect the dynamics of the body It is interesting to observe that the reduced equation (21)d , stating the moment equilibrium around the tangent, can be obtained in a much simpler way by substituting reduced expressions of the strains (4) and (8), based on the estimates performed here, in the linear part of the equilibrium equation (12)d However, this is a posteriori observation, emerging from the analysis, but unpredictable a priori Equations (21) must be integrated with the reduced boundary conditions: u = v = w = 0, GJ(ϑ′ + κ ¯ w ′ + w ′ v ′′ ) = 0, at s = 0, l (22) Equations expressing the vanishing of the bending moments must instead be ignored, consistently with the approximation adopted, that does not permit to describe the boundary layers 3.2 Static condensation It is well known that, in the framework of the parabolic cable theory, the tangential inertia force mă u can be neglected in the prevalently transversal motions, since the longitudinal natural frequencies are much higher than the transversal ones This circumstance permits to statically condense the tangent displacement u, by expressing it as an integral of the transverse displacements v and w: s u(s, t) = e(t)s + 1 [¯ κv(ξ, t) − v ′ (ξ, t)2 − w ′ (ξ, t)2]dξ 2 where e(t) = − l l 1 [¯ κv − v ′2 − w ′2 ]ds 2 (23) (24) An analogous procedure is here applied to the equation (21)d governing the twist Since the torsional frequencies of a single cable turn out to be much higher than the transversal ones, the inertia couple J1 ă is neglected and obtained in integral form To solve (21)d and the boundary conditions (22)d in the unknown ϑ, first v and w are scaled by a perturbaton parameter ǫ ≪ 1, then ϑ is expanded in ǫ-series: v → ǫv, ϑ = ǫϑ1 + ǫ2 ϑ2 + w → ǫw, (25) The following perturbation equation are obtained: ǫ: ǫ : GJϑ′′1 − EI κ ¯ ϑ1 = −¯ κ (GJ + EI) w ′′ GJϑ′′2 − EI κ ¯ ϑ2 = EI κ ¯ v ′′ ϑ1 − GJ(v ′′ w ′)′ (26) where c1 = has been taken for semplicity The relevant boundary conditions read: ǫ: ǫ2 : GJ (ϑ′1 + κ ¯w′) = GJ (ϑ′2 + v ′′ w ′ ) = (27) By solving in chain Eqs (26), it follows: GJ + EI s ′′ w (ξ, t) sinh [k(s − ξ)] dξ + A1 cosh ks + B1 sinh ks ϑ1 (s, t) = − √ GJ EI EI s ′′ v (ζ, t)ϑ1(ζ, t) sinh [k(s − ζ)] dζ+ ϑ2 (s, t) = GJ s ′′ − (v (ξ, t)w ′(ξ, t))′ sinh [k(s − ξ)] dξ + A2 cosh ks + B2 sinh ks k (28) where k := κ ¯ EI/GJ has been set and where the arbitrary constants A1 , B1 , A2 , B2 are determined by Eqs (27) Aerodynamic forces Modelling aerodynamic loads is a very difficult task, often handled in literature under strong hypoteses Moreover, the iced cable problem adds further difficulties to the popular problem of indefinite cylinder, due to the curvature of its centerline and the random variation of the section To tentatively tackle the problem, a simple model is adopted here, by introducing the following assumptions: (a) the quasi-static theory [11] is believed applicable, according to which the loads acting on the moving body at a certain instant are identical to those exerted on the body at rest in the same position; (b) the curvature of the cable, due to its smallness, is negligible; (c) loads are evaluated in the current configuration C, by accounting for the twist angle ϑ, but neglecting the smaller flexural rotations ϑ2,3 = O (ϑd/l) (remember Eqs (3) and (18)), which, according to the so-called cosine rule [12], have small influence; (d) the ice is assumed ¯2 a20 a2 a −y(s) ay C¯ ϑϕ ax C0 ¯i a s az (a) V ϕ ai0 U G a30 ≡ az ϕ ¯3 a ϑ γ a3 (b) bd u˙ U bl Figure 2: Aerodynamic forces: (a) cable configuration; (b) transversal section, mean wind velocity U, relative wind velocity V, angle of attack γ, drag force bd and lift force bl to be uniformly distributed along the cable, consistently with the hypotesis of planar reference configuration; (e) the aerodynamic couples are neglected Let us now consider a wind flow of mean velocity U = Uaz , blowing horizontally and normally to the initial (no wind) planar configuration of the cable (Fig 2a ) Three different attitudes of the cross-section in its own plane are considered (Fig 2b ): (a) the initial configuration C0 (axes a20 , a30 ≡ az ), in which the cable is only subjected ¯2 , a ¯3 ), in which the cable is loaded to gravity; (b) the reference configuration C¯ (axes a also by (uniform) static wind forces; (c) the actual configuration C (axes a2 , a3 ), in which the cable is loaded also by (non uniform) dynamic wind forces The twist angle caused by the static forces coincides, to within small quantities of order O(d2 /l2 ), with the angle of rotation ϕ experienced by the cable in passing from C0 to C¯ (see Fig 2a,b ), which depends only on the mean wind velocity U The twist angle ϑ caused by the dynamic forces depends, in addition to U, on the abscissa s and on time t The angle ϕ is assumed to be large; the angle ϑ is assumed to be small but finite According to the quasi-static theory, the flow exerts on the section the following aerodynamic force: ba = ρV r(cd (γ)V + cl (γ)a1 × V) (29) where ρ is the air density, V is the relative velocity of the wind respect to the section, V = ||V|| its modulus, and cd and cl two aerodynamic coefficients, called of drag and lift, respectively These latter depend on the shape of the section and on the angle of attack, V · a2 (30) γ := − arcsin V i.e on the angle between V and a reference material axis, here taken as a3 The two components of ba , along-wind bd and cross-wind bl , are usually known as drag and lift forces, respectively (Fig 2b ) 10 Equations (29) and (30) permit to evaluate the force ba once the relative velocity V is known If the section underwent only a translation va2 + wa3 , the relative velocity would be easily obtained as V = U − v¯ ˙ a2 − w¯ ˙ a3 In contrast, a non-vanishing twist ˙ velocity ϑ entails some difficulties, since V becomes a function of the point P on the boundary To overcome the problem, the notion of characteristic radius [11] has been introduced in literature, which consists in selecting a special point Pc on the boundary of the section, in which to evaluate a “characteristic relative velocity” Vc , to be attributed to all the points of the section The problem at hand, however, is simpler In fact, the contribution of ϑ˙ to the velocity of any point on the boundary of ˙ the section is of order O(ϑr), r being a transversal dimension Since, by Eq (18)b , ˙ ϑ ≤ O (w/d), ˙ the previous contribution is at the most of order O (wr/d), ˙ and therefore ˙ it is negligible respect to w˙ (on v) ˙ Hence, the twist velocity ϑ has practically no effects on the aerodynamic forces of cables having not evanescent sag In contrast, the twist angle ϑ does affect the forces via the angle of attack γ; conseguently ba = ba (ϑ, v, ˙ w; ˙ ϕ(U), U) By letting V = U − v¯ ˙ a2 − w¯ ˙ a3 in Eqs (29) and (30) and expanding them for small v, ˙ w˙ and ϑ, it follows: V = U[1 − v˙ w˙ sin ϕ − cos ϕ)] + h.o.t U U (31) and w˙ v˙ cos ϕ − sin ϕ + h.o.t (32) U U ¯ i.e when (ϑ, v, Equation (32) shows that when C → C, ˙ w) ˙ → 0, then γ → −ϕ Hence, by expanding the aerodynamic coefficients cα (γ) (α = d, l) around γ = −ϕ, one has: α = d, l (33) cα (γ) = c¯α + (γ + ϕ)¯ c′α + (γ + ϕ)2 c¯′′α + ¯ Finally, by where c¯α , c¯′α , , are the values assumed by cα and its derivatives at C ¯2 and a ¯3 axes, the substituting Eqs (31)-(33) into Eq (29), and projecting it on the a following force components are derived: γ = −ϕ − ϑ + bai = ¯bai (ϕ)+ cij (ϕ)ξj + j=1 cijk (ϕ)ξj ξk + j,k=1 cijkl (ϕ)ξj ξk ξl i = 2, (34) j,k,l=1 where ξ := (ϑ, v, ˙ w) ˙ T is the vector collecting the independent variables In (34) ¯bai are the static forces, and cij , cijk and cijkl are coefficients depending on cd , cl and their ¯ Some of them are reported in Appendix C derivatives respect to γ, all evaluated at C Equilibrium path and stability ¯ in which it Let us consider the cable in the equilibrium reference configuration C, is loaded by its own weight −mgay , g being the gravity acceleration, and by the 11 ¯ a (ϕ, U) of the aerodynamic force Since, by hypotesis, C¯ is planar, steady-state part b ¯ ¯ a (ϕ, U) − mgay lies in the equilibrium requires that the resultant force b(ϕ, U) := b plane of the cable By vanishing the force component along the binormal direction, ¯ ¯3 = − sin ϕ, it follows i.e enforcing b(ϕ, U) · a3 = 0, since ay · a sin ϕ = − ¯ba (ϕ, U) mg (35) Equation (35) implicitly defines the nonlinear, non-trivial equilibrium path ϕ = ϕ(U) The stability of the equilibrium of a generic point of this path is governed by the incremental equations of motion (21) linearized around the reference configuration, together with the linearized boundary conditions (22) By accounting for Eqs (23), (24) and (28)a to express the linear tangential displacement and twist angle, and for Eq (34) to express the (linear) aerodynamic forces, it follows that: l κ ¯ T¯ v ′′ − EA vds + c21 ϑ + c22 v˙ + c23 w = mă v l T w ′′ + c31 ϑ + c32 v˙ + c33 w˙ = mwă (36) GJ + EI s w (, t) sinh [k(s − ξ)] dξ + A cosh ks + B sinh ks ϑ(s, t) = − √ GJ EI v = 0, w = 0, ϑ′ + κ ¯ w ′ = 0, in s = 0, l where the coefficients c22 and c33 are modified respect to Eq (34) to include the structural damping All the coefficents cij are wind velocity-dependent Equations (36) represents a differential linear eigenvalue problem, of non self-adjoint type, due to the presence of dissipative (velocity-dependent) and circulatory (position-dependent) iλt forces It admits infinite solutions of type v(s, t) = vˆ(s)eiλt , w(s, t) = w(s)e ˆ , where the eigenvalues λ depend on the velocity U For small U, ℜ(λ) < for any λ, so that C¯ is a stable equilibrium configuration At a critical wind velocity Uc , however, the couple of complex conjugate eigenvalues having maximun real part crosses the imaginary axes, i.e max ℜ(λ) = This circumstance causes loss of stability of the equilibrium through a Hopf bifurcation, from which a limit cycle arise, of stable (supercritical, U ≥ Uc ) or unstable (undercritical, U ≤ Uc ) kind Evaluation of the postcritical behavior calls for using the whole nonlinear equations of motion (21), along the lines of previous papers by authors [1, 8], but it is out of the scope of the present work Therefore, analysis is so far confined to evaluation of the condition of incipient instability (U = Uc ) To solve the differential boundary problem (36), a Galerkin approach is followed, in which the in-plane (φvj ) and out-of-plane (φwk , φϑk ) eigenfunctions of the associated Hamiltonian problem (cij = 0) are taken as trial functions, namely:       n m ϑ(s, t) φ θk  v(s, t)  = φvj  qji +   qko (37) j=1 k=1 w(s, t) φw k 12 where qji , (j = 1, , m), are the unknown amplitudes for the in-plane trial functions and qko , (k = 1, , n), are the unknown amplitudes for the out-plane trial functions By using standard methods, the following algebraic eigenvalue problem is obtained: Mă q + Cq + (K + H) q = (38) where q = (qji , qko ) is the m + n-vector of the Lagrangian parameters, and M, C, K and H are the mass, damping (structural plus aerodynamic), stiffness and circulatory matrices, respectively These are found to be block-diagonal, due to symmetricantisymmetric character of the eigenfunctions Their coefficients are reported in Appendix D for m = n = Numerical results A preliminary numerical analysis has been performed on a sample cable, already analyzed in the literature (e.g [1]), having an axial rigidity EA of 29.7×106 N, a diameter of 0.0281 m, a length l of 267 m, a sag d of 6.18 m and damping ratio coefficients equal to 0.44% According to these values, the cable is initially close to the first cross-over point (Irvine & Caughey [10]) Two different U-shaped conductors are taken into account, always referring to literature: a cross-section with the symmetry axis placed on az direction (cross-section1 in the sequel), having its maximum ice eccentricity facing the wind (m = 1.80 kg/m, ice included; Yu et al [4]), and a cross-section with the symmetry axis rotated of −44.4◦ respect to az -direction (cross-section2 in the sequel), having greater ice thickness (m = 2.00 kg/m ice included; Tunstall [13]) In both the cases the specified configuration is the more prone to galloping It should be noted that, in the proposed theory, this position corresponds to no-wind conditions since the angle of attack γ is also statically varying through the angle of rotation ϕ (see Eq (32)) Therefore, when galloping actually occurs, the cable cross-section is rotated from the more dangerous initial position At the lower level, the displacement field v(s, t), w(s, t) and ϑ(s, t) can be approximated by the first symmetric in-plane φv (s) and out-of-plane (φw (s), φϑ (s)) eigenfunctions of the corresponding Hamiltonian system Figure shows the nonlinear equilibrium path ϕ = ϕ(U) (Fig 3a ) for the two different cross-sections in the basic case, and the changes in the pre-stress T¯ due to the static loads (Fig 3b ) Differences are due to aerodynamic coefficients and to cable mass When the mean wind velocity increases, the rotation soon achieves relevant values and the pre-stress is subjected to non-negligible alterations The conditions of incipient instability are examined evaluating the real part of the two couples of complex conjugate eigenvalues for the discretized system The objective is to point out the possible role of the dynamic twist angle ϑ on the critical wind velocity Uc Concerning the basic case for both the cross-sections, Figure shows the differences in the real part of the critical eigenvalue considering (continuous lines) or neglecting (dashed lines) the circulatory matrix H, 13 described in the Appendix D About the cross-section1 (Fig 4a ), differences are limited to the second bifurcation point B2 , where the stability of the planar equilibrium configuration is regained Concerning the cross-section2 (Fig 4b ), the circulatory matrix has quantitatively small influence, but it is decisive on the occurrence or not of bifurcations 30000 (a) (b) 28000 T (N) ϕ (rad) -0.2 26000 -0.4 -0.6 12 16 24000 20 12 16 20 U (m/s) U (m/s) Figure 3: (a) Nonlinear equilibrium path; (b) Pre-stress (thick lines: cross-section1; thin lines: cross-section2) 0.02 0.01 B1 B2 B1 -0.02 ℜ(λ) ℜ(λ) -0.04 -0.01 -0.02 -0.06 -0.03 (b) (a) -0.08 B2 10 -0.04 15 U (m/s) 10 15 U (m/s) Figure 4: Real part of the critical eigenvalue: (a) cross-section1; (b) cross-section2 (continuous lines: complete model; dashed lines: circulatory matrix H neglected) It was previously found that the dynamic twist ϑ is much higher in symmetric modes than in antisymmetric ones In order to quantify these results, the coefficients of the matrix H obtained by the Galerkin procedure for m = n = 1, and evaluated for symmetric and antisymmetric modes, are compared (Fig 5) Three values of d are considered to explore the situation of almost slack (d = 27 m) and almost taut (d = m) cable beside the basic case, maintaining the original length l = 267 m The effective influence of the dynamic torsion seems always negligible for the antisymmetric 14 0.6 1st symmetric mode 1st antisymmetric mode st symmetric mode 1st antisymmetric mode 0.4 h22 m22 h12 m11 0.2 d1 d3 d2 d3 -0.2 d2 d1 -0.4 (a) -1 (b) 12 16 -0.6 20 12 16 20 U (m/s) U (m/s) Figure 5: Coefficients of the circulatory matrix H: (a) coupling terms; (b) terms modifying the structural stiffness (d1,2,3 = 3, 6.18, 27 m; l = 267 m; cross-section1) modes, even for high values of the sag On the contrary, the dynamic torsion appears remarkable as regards symmetric modes, especially when the curvature (i.e the sag) decreases To better investigate this aspect with its implications on the system stability, the previous examples related to symmetric modes are reconsidered with suitable changes in mechanical and aerodynamic parameters Concerning the cross-section1, a reduction of sag and an increase of damping cause that a cable, that is unstable not considering the dynamic torsion, is actually stable (Fig 6a ) Moreover, starting the analysis with a cross-section slightly rotated (e.g −1◦ ) as regards the position more prone to galloping, the cable cross-section reaches the more dangerous attitude for instability in an inclined equilibrium configuration, then in proximity of bifurcation points In this way, bifurcations in both the solutions exist and appreciable differences are obtained (Fig 6b ) Alteration even larger can be obtained with further increasing in damping ratio In these latest examples, the contribution of dynamic torsion improves the system stability, but this condition does not appear as a general rule The influence of dynamic torsion seems still more pronounced on the cross-section2 Maintaining the basic sag and considering an initial rotation of the cross-section equal to −47◦ (instead of the basic value of −44.4◦ ) differences between the two bifurcation points are straightaway found (Fig 7a ) If the role of the torsion is exalted decreasing the sag (Fig 7b ), large alterations of the critical wind velocities appear Conclusions The aim of this paper concerns the formulation of a consistent cable-beam model able to considering the twist angle, which can be very important in the determination of aeroelastic behavior of this kind of structures Several points are worth highlighting A consistent model of a nonlinear, curved, prestressed, no-shear, elastic cable15 0.02 0.02 B1 B1 B2 B2 ℜ(λ) ℜ(λ) -0.02 -0.04 -0.02 -0.04 -0.06 -0.06 -0.08 -0.1 (a) (b) 10 -0.08 15 U (m/s) 10 15 U (m/s) Figure 6: Real part of the critical eigenvalue of the cross-section1: (a) d = m; (b) d = m and sectional symmetry axis rotated of −1◦ in no-wind conditions (damping coefficients equal to 0.65%; continuous lines: complete model; dashed lines: matrix H neglected) beam has been formulated Reduced equations of motion have been deducted through a suitable magnitude order analysis; this has made possible to clarify the different role of the dynamic twist angle on symmetrical and antisymmetrical modes; as major result, the reduced equations of motion are identical to those of a flexible cable, with an additional nonlinear equation in the twist angle The dynamic twist angle is a passive variable, slave of translations The aerodynamic forces have been evaluated taking into account the angle of static rotation induced by the mean wind; the corresponding change in the configuration of the cable has been considered The effective importance of the twist velocity on the aerodynamic forces has been discussed for cables having not evanescent sag Numerical results are preliminarily obtained as regards the linearized reduced equations of motion, using a Galerkin procedure with translational and twist eigenfunctions, in order to study conditions of incipient instability It has been proved that the dynamic twist angle is able to sensibly influence the critical conditions of the system, through the circulatory matrix, when symmetrical modes are taken into account, especially for small values of sag The presence of twist angle may imply the appearance or disappearance of criticality, and may lead to remarkable differences on aeroelastic critical velocities These alterations seem more pronounced when a cross-section in an initially non-symmetric position is considered 16 0.02 0.008 (b) (a) 0.004 B1 ℜ(λ) ℜ(λ) 0.01 B2 B1 -0.004 B2 -0.008 -0.01 -0.012 -0.016 10 15 -0.02 10 15 U (m/s) U (m/s) Figure 7: Real part of the critical eigenvalue of the cross-section2: (a) d = 6.18 m; (b) d = m (sectional symmetry axis rotated of −47◦ in no-wind conditions; continuous lines: complete model; dashed lines: matrix H neglected) A Bending and torsion of a curve beam By differentiating Eq (1)b and using Eq (6)b , it follows: ¯j + R¯ ¯j a′j = R′ a κ×a (39) By letting κ = 3i=1 κi , Eq (6)a furnishes: κi = a′j · ak , where i = j = k are permutations of 1, 2, By accounting for Eqs (1)b and (39), one has: ¯j · a ¯k + κ ¯ ·a ¯i κi = a′j · R¯ ak = RT R′ a (40) where use of definition of the transpose tensor, of RT R = I and permutation of the ¯i , Eq (5) and (40) ˆ = 3i=1 κ ˆia mixed vectorial product has been made By defining κ lead to: ¯j · a ¯k κ ˆ i = RT R′ a (41) Hence, κ ˆ i are the component of the axial vector of tensor RT R′ (Eq (7)) B Magnitude order of the twist angle The linearized equation (21)d is considered, in which c1 = and EI = GJ is assumed Moreover, w= cos nπs l nπs sin l n = 1, 3, n = 2, 4, s ∈ − 2l , 2l (42) is taken, for symmetric and antisymmetric modes, respectively The equation admits the following appoximate solution: ϑ≃ A − 2¯ κ cos nπs l B¯ κs − 2¯ κ sin nπs l 17 n = 1, 3, n = 2, 4, (43) since κ ¯ l ≪ and therefore cosh κ ¯ s ≃ and sinh κ ¯s ≃ κ ¯ s in the interval of interest ′ ′ By requiring ϑ + κ ¯ w = at s = ±l/2, the arbitrary constants are found to be A = ±2nπ/(¯ κl2 ), B = ±nπ/l, from which, for n small: ϑ= O O w d wd l2 n = 1, 3, n = 2, 4, (44) Using (44), the first and second derivates of ϑ are found as in Eq (20), both for symmetric and antisymmetric modes By summarizing, since the symmetric mode is slowly-varying, in order to satisfy boundary conditions the amplitude A must be large; on the other hand, since the antisymmetric mode is fastly-varying, the amplitude B must, in contrast, be small C Static wind forces and aerodynamic coefficients The static wind force components ¯bai and the coefficients c’s expressing the dynamic wind force components bai − ¯bai , all appearing in Eqs (34), are listed below: ¯ba = ρU r (−cl cos ϕ + cd sin ϕ) 2 c21 = ρU r (c′l cos ϕ − c′d sin ϕ) c22 = ρUr (−3cd − c′l + (cd − c′l ) cos 2ϕ + (c′d + cl ) sin 2ϕ) c23 = ρUr (−c′d + 3cl + (c′d + cl ) cos 2ϕ − (cd − c′l ) sin 2ϕ) ¯ba = ρU r (cd cos ϕ + cl sin ϕ) c31 = − ρU r (c′d cos ϕ + c′l sin ϕ) c32 = − ρUr (−c′d + 3cl − (c′d + cl ) cos 2ϕ + (cd − c′l ) sin 2ϕ) c33 = − ρUr (3cd + c′l + (cd − c′l ) cos 2ϕ + (c′d + cl ) sin 2ϕ) (45) Higher-order coefficients are not reported here for brevity D Coefficients of the algebraic eigenvalue problem (38) When m = n = 1, the langrangian parameters vector is q = (q1i , q1o ) The mass matrix is: m11 (46) M= m22 18 where l l φ2v1 ds, m11 = −m φ2w1 ds m22 = −m (47) The structural stiffness matrix is: K= k11 0 k22 (48) where l k11 = T¯ k22 = T¯ φv1 φ′′v1 ds − l EA¯ κ2 l l φv1 ds φv1 ds 0 l l φw1 φ′′w1 ds φϑ1 φ′′ϑ1 ds + GJ l φ′′w1 φϑ1 ds+ +κ ¯ (GJ + EI) (49) l − EI κ ¯2 φ2ϑ1 ds The circulatory matrix is H= h12 h22 (50) where: l h12 = c21 l φv1 φθ1 ds, h22 = c31 φw1 φθ1 ds (51) The damping matrix is C = Cs + Ca , where Cs is the structural damping matrix: √ 2ξ1 m11 k11 √0 Cs = 2ξ1 m22 k22 (52) and ξ1 , ξ2 are the modal damping factors, while Ca is the aerodynamic damping matrix: c c Ca = a11 a12 (53) ca21 ca22 where l l φ2v1 ds, ca11 = c22 ca12 = c23 ca21 = c32 φv1 φw1 ds, l l φv1 φw1 ds φ2w1 ds ca22 = c33 0 Acknowledgements This work has been partially supported by a MIUR-2005 grant 19 (54) References [1] A Luongo, G Piccardo, “Non-linear galloping of sagged cables in 1:2 internal resonance”, Journal of Sound and Vibration, 214(5), 915-940, 1998 [2] A Luongo , G Rega, F Vestroni, “Planar non-linear free vibrations of an elastic cable”, International Journal of Non-Linear Mechanics, 19(1), 39-52, 1984 [3] C.L Lee, N.C Perkins, “Nonlinear oscillations of suspended cables containing a two-to-one internal resonance”, Nonlinear Dynamics, 3, 465-490, 1992 [4] P Yu , Y.M Desai, A.H Shah, N Popplewell, “Three-degree-of-freedom model for galloping Part I: formulation; Part II: solutions”, Journal of Engineering Mechanics, ASCE, 119(12), 2404-2448, 1993 [5] A Luongo, G Piccardo, “On the influence of the torsional stiffness on nonlinear galloping of suspended cables”, Proceedings, 2nd ENOC, Prague, Czech Republic, 1, 273-276, 1996 [6] G Diana , S Bruni , F Cheli , F Fossati, A Manenti, “Dynamic analysis of the transmission line crossing Lago de Maracaibo”, Journal of Wind Engrg and Ind Aerodyn., 74-76, 977-986, 1998 [7] A Luongo , D Zulli, G Piccardo, “A linear model of curved beam for the analysis of galloping oscillations in suspended cables”, Proc., XVI AIMETA Conference, Florence (Italy), CD-Rom, 2005 (in Italian) [8] A Luongo , A Paolone, G Piccardo, “Postcritical behavior of cables undergoing two simultaneous galloping modes”, Meccanica, 33, 229-242, 1998 [9] S Wolfram, “The Mathematica book”, Cambridge University Press, 2005 [10] H.M Irvine, T.K Caughey, “The linear theory of free vibrations of a suspended cable”, Proceedings of the Royal Society of London, A341, 299-315, 1974 [11] R.D Blevins “Flow-induced Vibration”, Van Nostrand Reinhold, second edition, New York, 1990 [12] E Strømmen, E Hjorth-Hansen, “The buffeting wind loading of structural members at an arbitrary attitude in the flow”, Journal of Wind Engineering and Industrial Aerodynamics, 56, 267-290, 1995 [13] M Tunstall, “Accretion of ice and aerodynamic coefficients”, Proceedings of the Association des Ing´enieurs Montefiore AIM, Study Day on Galloping, University of Liege, Liege, Belgium, 1989 20 ... equations of motion are identical to those of a flexible cable, with an additional nonlinear equation in the twist angle The dynamic twist angle is a passive variable, slave of translations The... twist is a passive variable, slave of the normal and binormal translations The reduced model thus obtained permits to investigate the critical and postcritical aeroelastic behavior of the cable,... coupling Therefore, the formulation of a consistent cable -beam model is a matter of great interest, able to take into account all the stiffnesses involved in the problem To the best of authors’ knowledge,