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An experimental and analytical study of autoparametric resonance in a 3DOF model of cable-stayed-beam

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An experimental and analytical study of autoparametric resonance in a 3DOF model of cable-stayed-beam
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  An Experimental and Analytical Study of Autoparametric Resonance in a 3DOF Model of Cable-Stayed-Beam Y. FUJINO Department of Civil Engineering, University of Tokyo, Tokyo 113, Japan P. WARNITCHAI Division of Structural Engineering and Construction, Asian Institute of Technology, Bangkok, Thailand B. M. PACHECO Ammann and Whitney INC., New York, NY 10014-3309, U.S.A. (Received: 15 August 1990; accepted: 21 April 1992) Abstract. Autoparametric interaction and the associated phenomenon of amplitude saturation are experimentally observed in a physical model of cable-and-beam structure. In this system, the horizontal beam is fixed at one end and supported at the other end by an inclined taut cable, The longitudinal axes of beam and cable are in a vertical plane. Three natural frequencies of the system are approximately of the ratio 1 : 1 : 2. This is a combination of two conditions that are very likely to occur in relatively long-span, multi-stay-cable bridges, namely, l : 1 tuning and 1 : 2 superharmonic tuning. While the beam is vertically excited with sufficiently large force near a primary resonance, the cable vibrates horizontally at half of excitation frequency. The beam also vibrates horizontally at half-frequency, as well as vertically. As the vertical excitation on the bean is further increased in amplitude, the vertical vibration amplitude gets saturated instead of increasing proportionately. A 3DOF analytical model of the structure is also derived, where the finite motion of the cable introduces geometric nonlinearities in quadratic and cubic forms. The system parameters having been carefully measured from the experimental model, steady-state solutions of the coupled nonlinear equations of motion are obtained, by the perturbation method of multiple time scales. Agreement between experimental observation and analytical prediction is very good, both qualitatively and quantitatively. Very good agreement is found also in the case of horizontal excitation of the beam, where effects of linear and nonlinear interaction are apparent, Key words: Autoparametric resonance, saturation, cable-stayed bridge, geometric nonlinearity. 1. Vibration of Cable-Stayed Bridge-an Application From an engineering viewpoint, vibrations of cable-stayed bridges can be classified into two types, namely 'local' vibrations and 'global' vibrations [1]. It has been common practice in their dynamic analysis to treat these two types of vibration separately instead of considcring any interaction between them. Cable vibrations are local, in the sense that the anchorage points at girder and pylon are fixed. On the other hand, girder-pylon vibrations are global, since the whole bridge span vibrates. In the latter type of vibration, the cables do not vibrate locally but behave instead like massless elastic tendons. Separate treatment of local and global vibrations is not sufficient, however, when the respective natural frequencies of local and global vibrations become nearly equal to each other. In this case, some modes of the total structure become closely spaced in frequency (e.g., [2], [3], [4]). Nonlinear Dynamics 4: 111-138, 1993. © 1993 Kluwer Academic Publishers. Printed in the Netherlands  112 Y. FUJINO ET AL. Each of these modes may be viewed as strong coupling of local and global vibrations, judging by the modal shape. When amplitudes are large, i.e., the cables undergo finite motion, the coupling can also be nonlinear. Quadratic coupling will cause autoparametric interaction when two of the natural frequencies are in the ratio one to two. As discussed in the present study, such effects of geometric nonlinearity may be significant. As an essential simplification of cable-stayed bridges with stiff pylon (Figure l(a)), the structure considered herein (see also [5]) is a cantilevered beam that is supported at one end by an inclined taut cable (Figure l(b)). The secondary light wire that ties the cable serves to increase the natural frequencies in in-plane local cable vibrations. Hence local cable vibration in the vertical plane is not an important degree of freedom. For relatively long span, the natural frequency fv (also see Appendix- Nomenclature) of the cable in local horizontal vibration (Figure 2a) can be low enough to be almost equal to the natural frequency fh of beam global horizontal vibration (Figure 2(b)). This condition leads to the linear interaction. Note that cubic couplings which cause a nonlinear interaction are in fact negligibly small in this situation [5]. At the same time, fv can be nearly half the natural frequency fg of the beam in global vertical vibration. Figure 2(c) shows the second global vertical mode. This condition can lead to autoparametric interaction. These two conditions, f v : fh = 1 : 1 and fv : f~ = 1 : 2, are very likely in long-span, multi-stay-cable bridges, and they may occur simultaneously. For example, an actual pedestrian bridge has been reported that satisfies the condition fy : fh : fg = 1 : 1 : 2 between some stay cables and the girders [6]. When this happens, it is of particular interest to see show local and global vibrations interact, in the presence of vertical excitation on the beam that is nearly resonant with global vibration, i.e., the frequency of vertical excitation nearly equals fg. A major part of the present study is devoted to exploring the above phenomenon in a carefully designed experiment, using the model shown schematically in Figure l(b). A three- degree-of-freedom (3DOF) analytical model is also developed for quantitative verification of experimental results. (a) 9 Fig. l. (a) Cable-stayed bridge; (b) cable-stayed beam.  AUTOPARAMETRIC RESONANCE 113 z/z / (a) (b) l Fig. 2. Local and global vibration shapes, (a) cable (normal to the vertical plane), (b) beam horizontal. (c) beam vertical. 2. Analytical Model of Cable and Beam Vibration 2. 1. Nonlinear Equations of Motion The beam displacements are denoted as u(x, t) for the horizontal component and v(x, t) for the vertical. For the cable, u(s, t) denotes the horizontal component (normal to the vertical plane), while v(s, t) and w(s, t) stand for the components perpendicular and parallel to the cable axis, in the vertical plane, respectively. All the displacements, u(x, t), v(x, t), u(s, t), v(s, t) and w(s, t) are taken from the static equilibrium. Generalized coordinates ~y, ~b h, and 4~g (Figure 2) and the corresponding generalized displacements y, h, and g are used to describe vibrations of the cable-and-beam structure in terms of three degrees of freedom: u(x, t) qS,,(x)h(t) (1) v(x, t) = 4~x(x)g(t) (2) u(s, t) = fb,,(s)y(t) + 49,,(x.)h(t)s/L, (3) v(s, t) = 4~(x, )g(t)(cos O)s/L, (4) w(s, t) = eh~,(x, )g(t)(sin O )s/ L , (5) where chh(x) and ~b~,(x) are normalized global modes satisfying q~,,(L) = 1 ~b(L) = 1 and &v(s) = sin( rcs / L ,) ,  114 Y. FUJINO ET AL. i.e., the local mode ~bv(s is the normalized first transverse mode of a taut string with uniform mass per unit length, /~, and elastic modulus, EcA ~. The coordinates x and s are defined in Figure 2. The dimensions L, L c, xc, and 0 are defined in Figure l(b). The idealization of the cable as a string is justified by the fact that, for stay cables in actual bridges, the tension is larger than the unit weight by about two orders of magnitude. The omission of the longitudinal degree of freedom in the cable is also reasonable, since the elastic modulus is so high for metallic strings that the fundamental frequency of longitudinal mode is higher than the fundamental transverse frequency by at least one order of magnitude (e.g., pp. 487-491 of [7]). As may be noted in equations (3)-(4), quasi-static displacements are additionally induced on the cable because of the displacement of the cable anchorage at x = x c. Finally it may be verified in equation (4) that in-plane transverse local vibration of the cable has been excluded, as the frequencies involved have been considerably raised and detuned by the in-plane tie wire. By the above definitions, y is the local vibration of the cable at midspan; h is the global horizontal vibration of the tip of the beam; and g is the global vertical vibration of the tip of the beam. 2.1.1. Formulation of Lagrangian The equations of motion in terms of y, h, and g are obtained by Lagrange's approach. Considering both cable and beam as linearly elastic, the potential energy, U, is evaluated: U = Ucabl e "[- Ubeam . (6) According to Washizu [8], the incremental energy of the cable from the static equilibrium can be expressed as the sum of the potential due to the initial (static) stresses associated with the initial elongation u0, and the potential due to dynamic stress (strain) (see equation (7)). This extract potential at static equilibrium, and need not to use total strain in the energy formulation. In metallic strings, longitudinal inertia is negligible because of high initial axial stress [7]. Hence the dynamic strain can be assumed to be uniform along the cable span, i.e., it depends only on time. Therefore, the cable energy can be expressed as: 1 uoE~A ~ foL~{(OU(S, t))2 (Or(S, t))2 (OW(S, t))2} 1 Ucab~¢ -- 2 L,. Os + \ Os + \ Os ds + ~ EcA~LceZ(t ) (7) e(l) + dgg(Xc)g(t~ sin 0 (8) The flexural potential of the beam, beam, Ubeam s: d2 h) 2 1 1 h2(t ) EI h (--~x2/ dx + ~ g2(t) EIg \ dx 2 / dx (9) Ubeam = 2 In equation (8), effect of quasi-static motion due to thg(Xc) is included. Note that only the nonlinear term of the dynamic strain, e(t) in equation (8) contributes to the incremental potential due to the initial stresses. It should be emphasized that consideration of finite displacement for the  AUTOPARAMETRIC RESONANCE ] 15 cable (equation (7)) will lead to geometric nonlinearities in the governing differential equations of motion. In equation (9), only the flexural potential due to h and g has been included; the effects of initial axial and flexural stress are negligibly small. EI h and Elg are the flexural moduli for horizontal and vertical bending, respectively. For the kinetic energy, T, the following expressions are straightforward: T = Tc.b, e + Tbeam (10) 1 fL~I OU S,t))~- OV S,t)) 20w s,t))~}d s (11) T~"b'°=2 /xc , t\ ~- + Ot + O~ 1 (dh)2f L 1 (dg]2f L Ybeam = ~ "~ ) tt(x)ch](x) dx + 5 \dt / ) tt(x)cb~(x) dx, (,12) where t~(x) is the mass per unit length of beam. 2.1.2. Nondimensionalization The following nondimensionalization is adopted, so that the relative magnitudes of different terms in the differential equations can be ascertained systematically: ~ z y b/o g=h__ U 0 rr-uoE ~A ~ r=~ ~ t Lc #,, d2( ) 7r~-uoEcAc d2( ) ) .... d~ 2 Lcl~ ' dt 2 (13) (14) (15) (16) (17) For the present physical problem, the following relations of magnitudes are true: (1) the cable motion amplitude is much larger than the beam motion; (2) the initial elongation of the cable is small compared to its length; and (3) the generalized mass of the cable is much smaller than that of the beam. These three conditions: u n ~ L o My ~ M h M s where
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