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Non-linear coupled vertical and torsional vibrations of suspension bridges are investigated. Method of Multiple Scales, a perturbation technique, is applied to the equations to find approximate analytical solutions. The equations are not discretized as usually done, rather the perturbation method is applied directly to the partial differential equa...
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... three-span suspension bridge is used as a model. Each span is simply supported and a general view of a suspension bridge model is shown in Fig. 1. The non-linear equations of motion of coupled vertical-torsional vibrations were obtained by employing Hamilton's principle [1]. These equations were later used in another paper [6] by neglecting cross-sectional distortion, longitudinal inertia, shear deformation and rotatory inertia, and taking into account large deformation only ...
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... Figs. 9-19, a 10 and a 20 are, respectively, the initial vertical and torsional dimensionless amplitude values, 10 is the initial dimensionless value of the phase angle between the vertical and torsional frequencies, 20 is the initial dimensionless value of the phase an- gle between the vertical frequency and the external excitation frequency, 1 ...
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... mode is transferred to the torsional mode. These constant values are usually referred to as stationary or steady-state values. When the steady-state value of amplitude is non-zero, the free oscillation term is periodic. In this section, we investigate the effect of changing the initial amplitude and phase values on the behavior of the system. In Fig. 11, a 10 is taken as 0.02 and a 20 is still zero. In this case a 1 curve makes lesser oscillations when compared with that of Fig. 9, then it reaches the previous steady-state value. The a 2 curve, on the other hand, reaches the previous steady-state value in a shorter time. In Fig. 12, both a 10 and a 20 are taken as 0.02. In this case ...
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... amplitude and phase values on the behavior of the system. In Fig. 11, a 10 is taken as 0.02 and a 20 is still zero. In this case a 1 curve makes lesser oscillations when compared with that of Fig. 9, then it reaches the previous steady-state value. The a 2 curve, on the other hand, reaches the previous steady-state value in a shorter time. In Fig. 12, both a 10 and a 20 are taken as 0.02. In this case both curves reach the steady-state values after a shorter transition period. The simulation is performed with different initial conditions in Fig. 13; a 10 and a 20 are taken as 0.04 and 0.02, respectively. Both curves reach the same steady-state values after a transition period. ...
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... then it reaches the previous steady-state value. The a 2 curve, on the other hand, reaches the previous steady-state value in a shorter time. In Fig. 12, both a 10 and a 20 are taken as 0.02. In this case both curves reach the steady-state values after a shorter transition period. The simulation is performed with different initial conditions in Fig. 13; a 10 and a 20 are taken as 0.04 and 0.02, respectively. Both curves reach the same steady-state values after a transition period. Evaluating Figs. 9-13, one may notice that the steady-state values are independent of the initial values. The only effect of initial values is observed on the oscillations and duration of the transition ...
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... the oscillations and duration of the transition period. Whatever the initial amplitude and phase values are, the system reaches the same steady-state values; the closer the initial values to the steady-state values, the shorter is the transition period. On the other hand, the steady-state values are proportional to the energy level of the system. Fig. 14, the amplitude of excitation is increased to two times of that of Fig. 9. In this case, the forms of the curves remained almost the same whereas the steadystate value of a 1 has increased slightly and that of a 2 has decreased slightly. In Fig. 15, the excitation amplitude is decreased to one-third of that of Fig. 9; then, a 1 curve ...
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... On the other hand, the steady-state values are proportional to the energy level of the system. Fig. 14, the amplitude of excitation is increased to two times of that of Fig. 9. In this case, the forms of the curves remained almost the same whereas the steadystate value of a 1 has increased slightly and that of a 2 has decreased slightly. In Fig. 15, the excitation amplitude is decreased to one-third of that of Fig. 9; then, a 1 curve has reached a rather low steady-state value after a few oscillations and a 2 remained absolutely at zero. This proves that when the amplitude of vertical excitation remains under a certain value, the energy level of the system is insufficient to ...
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... is decreased to one-third of that of Fig. 9; then, a 1 curve has reached a rather low steady-state value after a few oscillations and a 2 remained absolutely at zero. This proves that when the amplitude of vertical excitation remains under a certain value, the energy level of the system is insufficient to cause any coupled torsional motion. In Fig. 16 we assumed no excitation (f = 0) and a very low damping. In this case the energy continues to be exchanged between the two modes but it is continuously dissipated. Fig. 17, the damping coefficient is decreased to one half of that of Fig. 9. In this case, the oscillations in the transition region have larger amplitudes but the ...
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... proves that when the amplitude of vertical excitation remains under a certain value, the energy level of the system is insufficient to cause any coupled torsional motion. In Fig. 16 we assumed no excitation (f = 0) and a very low damping. In this case the energy continues to be exchanged between the two modes but it is continuously dissipated. Fig. 17, the damping coefficient is decreased to one half of that of Fig. 9. In this case, the oscillations in the transition region have larger amplitudes but the steady-state values are almost the same. The damping coefficient values are doubled in Fig. 18. This time no torsional motion is observed; that is, the energy level was not ...
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... the energy continues to be exchanged between the two modes but it is continuously dissipated. Fig. 17, the damping coefficient is decreased to one half of that of Fig. 9. In this case, the oscillations in the transition region have larger amplitudes but the steady-state values are almost the same. The damping coefficient values are doubled in Fig. 18. This time no torsional motion is observed; that is, the energy level was not sufficient to cause coupled torsional motion. Finally, we assumed no damping in Fig. 19. According to the linear theory, the amplitudes should grow indefinitely in this case; however, we observe that as the amplitudes grow, the non-linearity limits the ...
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... Fig. 9. In this case, the oscillations in the transition region have larger amplitudes but the steady-state values are almost the same. The damping coefficient values are doubled in Fig. 18. This time no torsional motion is observed; that is, the energy level was not sufficient to cause coupled torsional motion. Finally, we assumed no damping in Fig. 19. According to the linear theory, the amplitudes should grow indefinitely in this case; however, we observe that as the amplitudes grow, the non-linearity limits the growth, resulting in limit cycle. Since the two modes are strongly coupled, the energy imparted to one of them is continuously exchanged between them during the ensuing ...
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Citations
... There are many studies on static or dynamic wind loads for suspension bridges. Nevertheless, most studies focus on linear problems [12][13][14][15]. For example, it was found in Ref. [14] that the frequencies of the fundamental symmetric and anti-symmetric modes are relatively independent of the girder stiffness. ...
... However, evidence needs to be supplied for such treatments. Recently, the flutter of suspension bridges was studied with the mixed effects of static and dynamic wind loads [5,15]. There have been no detailed studies of the vortex-induced vibrations for bridges with static wind loads. ...
... However, these accurate modes are extremely complex. If the sine functions replace the precise free vibration modes of suspension bridges, one can also obtain high-precision results [4,15,38,39]. Therefore, we assumed θ n = sin nπx, n = 1, 2, · · · , and then used the Galerkin method to truncate the first three terms to obtain ...
A low stiffness makes long-span suspension bridges sensitive to loads, and this sensitivity is particularly significant for wind-induced nonlinear vibrations. In the present paper, nonlinear vibrations of suspension bridges under the combined effects of static and vortex-induced loads are explored using the nonlinear partial differential–integral equation that models the plane bending motion of suspension bridges. First, we discretized the differential–integral equation through the Galerkin method to obtain the nonlinear ordinary differential equation that describes the vortex-induced vibrations of the bridges at the first-order symmetric bending mode. Then, the approximate analytical solution of the ordinary differential equation was obtained using the multiple scales method. Finally, the analytical solution was applied to reveal the relationships between the vibration amplitude and other parameters, such as the static wind load, the frequency of dynamic load, structural stiffness, and damping. The results show that the static wind load slightly impacts the bridge’s vibrations if its influence on the natural frequency of bridges is ignored. However, the bridge’s vibrations are sensitive to the load frequency, structural stiffness, and damping. The vibration amplitude, as a result, may dramatically increase if the three parameters decrease.
... To the authors' knowledge, most of the existing studies of the non linear dynamic analysis mainly focused on the classical configuration of suspension bridge with cables and hangers in a vertical plane. For such kind of configuration, the cables and deck are fully coupled and condensed into one object in the governing function with ignoring the swaying motion of main cables [9]. However, in recent years, more and more suspension bridges, whose main cables have spatial (outward and inward types) geometric layout, were developed [10]. ...
Torsional vibration of long span suspension bridge can be commonly induced by various sources, and detailed investigation is required from both design and safety point of view. This study proposes a continuum model of the generalized suspension bridge using the Hamilton principle. This model can consider any inclination angle of main cables. The horizontal and vertical motions of main cables, and torsional motion of deck are considered individually in this model. The proposed model is verified by comparing its modal properties with those from a finite element model. The nonlinear primary resonances of the two lowest torsional modes are studied through multiple-scale method (MSM) under harmonic excitation. The effects of system parameters, such as inclination angle of hangers, sag-to-span ratio, tensile stiffness of main cable and torsional stiffness of deck, on the resonance responses are investigated in this study. Results show that, increasing inclination angle of main cable and torsional stiffness of deck can intensify the softening feature of the system. In contrast, increasing the sag-to-span ratio and tensile stiffness of main cable can strengthen the hardening feature.
... Some of them investigated the non-linear free and forced vertical-torsional coupling with the multiple scale method (MSM). [27][28][29][30][31][32] Capsoni et al. 33 investigated the torsional aero-elasticity and parametric resonance instabilities of suspension bridges via the MSM based on a non-linear continuum model. In these studies, although the full bridge models were proposed or adopted, most of them assumed the vibration of main cables only in the vertical plane. ...
This study proposes an approach to set up a continuum full bridge model with spatially inclined cables based on the Hamilton principle. The dynamic governing functions, considering the geometric non-linearities of cables and deck, represent simultaneously the vertical motion of deck and vertical–horizontal motion of cable. With the comparison of the modal properties obtained from the model to those from the accurate model, results show that the proposed model is capable of accurately simulating the modal properties. The primary resonance responses and corresponding frequency-response curves are obtained through the multiple-scale-method. A finite element (FE) model is established, and the corresponding non-linear dynamic analysis in time domain is conducted. Comparing the results from two models, it can be checked that the proposed model is reliable. According to the results of the proposed model, it is found that the second-order shape functions (SOSFs) play a significant role in the system response. Once the non-linear vibration of the bridge becomes significant only considering the excited mode with using the classical Galerkin decomposition cannot correctly predict the structure response. The SOSFs can be classified into stationary and vibrating components. The vibrating component can deviate the time-series of response from the harmonic wave, and the stationary component directly determines the mean value of the time-series.
... The complexity of this behavior has drawn much research attention on its nonlinear characteristics. Most existing studies [32][33][34] are based on the modal discretization modeling, where only one or two modes can be included in the analysis. This poses a problem of accuracy with the limited modes studied. ...
Nonlinear vibration analysis of complex structures is still a challenging topic due to limitations of existing algorithms. This paper proposes a highly efficient method to address this problem where the structure has only a portion of components with geometric or material nonlinearity under service load. Incremental harmonic balance (IHB) method serves as the main tool. The linear and nonlinear parts of the structure are separated with the introduction of interface forces at the boundaries. The analysis of the linear and nonlinear parts is interacting via the interface forces with the compatibility condition. Two numerical examples are analyzed for illustration of the method, i.e. a frame with base isolation system and a suspension bridge model. Various resonances are noted in the vibration of the structures suggesting the need of analyzing the nonlinear vibration of any structure with nonlinear components. The accuracy of the proposed method is validated with comparison the obtained results with that by Runge-Kutta method. Results show that the proposed method provides a neat and efficient tool for this purpose, when the nonlinear components are far less than the linear components in the system.
... Reference to [3] shows that the method of multiple scales has been applied to a wide variety of problems in physics, engineering, and applied mathematics. Moreover, it is very efficient for studying the problems resulting in modal interaction in dynamical and struc-tural systems subjected to different cases of internal and combinational resonances [12,13], including suspension combined structures [14] and cable-supported bridges [8,[15][16][17][18]. ...
... For this purpose, damping terms proportional to the first-order time derivative in displacements have been added to Eqs. (4) and (5) in [16], as it is traditionally accepted in dynamics of structures [28]. The similar approach was realized 10 years later in [17] with the only difference that the method of derivative expansions has been applied directly to the governing equations of suspension bridge motion without preliminary expansion in terms of eigenmodes. ...
The history of formulation of the efficient method for studying the nonlinear dynamic response of structures, damping features of which depend on natural frequencies of vibrations, is presented. This technique is the modified version of the method of multiple scales, the efficiency of which is illustrated by the examples of nonlinear vibrations of suspension bridges and plates subjected to the conditions of the internal resonances.
... It was shown that the influence of the higher order modes, when the suspension bridge is subjected to wind loading, is more significant than in the case when the bridge is subjected to a moving load. Besides the above studies on a linear system, the vibration based on non-linear models was also studied [11][12][13]. Ding [14,15] studied the periodic oscillations in a cablesupported bridge system under periodic external forces. Malik [16,17] formulated the nonlinear model of a cable-supported bridge structure from the principle of minimum potential energy to describing the behaviour of bridge deck with discussions on the solution stability criterion. ...
... Many authors applied these principles to study the vibrations response of suspension bridges. The authors of [21,22] used the continuous model proposed by Abdel-Ghaffar [11], and solved the system of equations by means of the multiple scale perturbative technique [23]. Recently, Arioli and Gazzola [24], trying to explain why torsional oscillations suddenly appeared before the Tacoma Narrows collapse, found out that, also in isolated systems, vertical oscillations may switch to torsional ones, as long as they become large enough. ...
... This requires the knowledge of the eigensolutions. [22]. It is worth noting that the symmetric modes include trigonometric and hyperbolic functions, conversely the skew-symmetric ones are represented by simple sinusoidal shapes. ...
... A recent work by Arioli and Gazzola [24], as previous ones by different authors [21,22] investigates internal parametric resonance potentially suffered by suspension bridges. In order to catch this phenomenon, it is necessary to consider the fully non-linear equation of motion, hereby for the first time supplemented by the aeroelastic operator. ...
The potential occurrence of internal parametric resonance phenomena has been recently indicated as a potential contributory cause of the appearance of critical dynamic states in long-span suspension bridges. At the same time, suspension bridges, in view of their flexibility, are prone to aeroelastic response, such as vortex shedding, torsional divergence and flutter. In this paper, a non-linear dynamic model of a suspension bridge is devised, with the purpose of providing a first attempt toward a unified framework for the study of aeroelastic and internal resonance instabilities. Inspired by the pioneering work of Herrmann and Hauger, the analyses have been based on a linearized formulation that is able to represent the main structural non-linear effects and the coupling given by aerodynamic forces. The results confirm that the interaction between aeroelastic effects and non-linear internal resonance leads to unstable conditions for wind speeds which can be lower than the critical threshold for standard aeroelastic predictions.
... Un certain nombre de modèles ont été développés pour des problématiques en lien avec les ouvrages d'art et plus particulièrement les ponts haubanés. Il s'agit la plupart du temps d'étudier la dynamique couplée entre un câble incliné et le tablier du pont, avec un faible nombre de degrés de liberté comme l'étude [Fujino et al., 1993], en prenant en compte les déplacements du support du câble pour l'article [Warnitchai et al., 1995], par l'utilisation des éléments finis dans les travaux [Gattulli et al., 2005] ou des modèles plus complexes dans l'étude [Çevik et Pakdemirli, 2005]. ...
L’application des résultats de la dynamique des systèmes non linéaires aux structures du Génie Civil a permis d’imaginer un contrôle de leurs vibrations. Il s’agit d’utiliser une structure auxiliaire couplée à la structure principale à contrôler. S’il est possible d’imaginer des structures auxiliaires de fortes masses, en couplant deux structures principales entre elles par exemple, la plupart du temps il s’agit d’une faible masse qui leur est couplée. Ces couplages peuvent être de nature différente et il est intéressant de séparer les couplages actifs et passifs. Si les couplages actifs sont très efficaces, ils nécessitent un apport d’informations et d’énergie, contrairement au cas passif. Les systèmes passifs les plus utilisés sont les Tuned Mass Damper (TMD), qui reposent sur un couplage linéaire entre les structures, principale et auxiliaire. Ils nécessitent d’être accordés en fréquence, ce qui pose des problèmes de durabilité. De récents travaux ont montré qu’il est possible d’utiliser un couplage non linéaire entre les structures, principale et auxiliaire, pour le contrôle passif, grâce au phénomène de localisation. De tels absorbeurs non linéaires sont nommés puits d’énergie non linéaire, ou Nonlinear Energy Sink (NES). Les travaux présentés dans ce document proposent des avancées sur deux points : d’un côté continuer à développer le rôle d’absorbeur de vibrations des NES et d’un autre côté explorer s’ils peuvent remplir d’autres rôles. Ainsi ce manuscrit se consacre à deux applications possibles des NES, comme absorbeurs de vibrations pour le contrôle vibratoire ou comme capteurs pour l’auscultation de l’état des structures. Dans le cadre du contrôle vibratoire, il s’agit d’homogénéiser une démarche d’analyse de comportement des structures couplées applicable à différents types de couplages. Ensuite il convient de déterminer si cette démarche est modifiable pour être appliquée dans d’autres cas, ici la prise en compte de la gravité et/ou de rhéologies plus complexes pour la structure principale. En se fondant sur les mêmes résultats d’analyse du comportement des structures couplées, une étude des NES pour l’auscultation de l’état des structures est menée. Il s’agit d’utiliser les propriétés des couplages non linéaires à basculer entre des comportements fortement différents pour de faibles variations des paramètres du système. Le principe d’un capteur pour l’auscultation de l’endommagement est ainsi décrit sur ces bases. Enfin ce manuscrit s’intéresse à la création d’un démonstrateur en échelle 1 — câble de 21 m pour 2,6 cm de diamètre — sur la base du contrôle des vibrations horizontales d’un câble par NES cubique et des vibrations verticales d’un câble par NES linéaire par morceaux.
... Nonlinear model includes geometric nonlinearities induced by the suspension cables and the high flexural-torsional slenderness of the deck, elastic instability phenomena full extensional-flexural-torsional coupling of suspension bridges when subjected to wind-induced excitations (Arena and Lacarbonara, 2012). Cevik and Pakdemirli (2005) investigated non-linear coupled vertical and torsional vibrations of suspension bridges. Three-dimensional nonlinear aerodynamic stability analysis and parametric analysis were taken to study the aerodynamic stability of the Runyang bridge over the Yangtze River (Zhang and Sun, 2004). ...
Until now, no available analytic solution considering suspender deformation was given to determine cable tension load at the middle of cable and suspender internal load of suspension bridge. Firstly, the classical theory without suspender deformation was taken to study the sag effect on the internal load of suspension bridge. Results indicate that the tension load at all cable points decreases when the sag increases. A procedure using Müller-Breslau principle to determine the moment and shear force influence line of the stiffening girder in the wire suspension bridge was given, and validated by the influence line given by the classical theory. Secondly, a new analytic solution considering suspender deformation was given to determine cable tension load at midpoint and the internal load of suspenders. The new analytic solution is more reasonable than the classical theory in theory. According to the new analytic solution, the cable tension load gets peak value when the unit live load Pi applies at the middle of stiffening girder, while the internal tension loads in all suspenders reach equal maximum value. Then a linear static model was built in ANSYS ignoring large displacement, initial strain and Ernst’s modulus of elasticity. Results using the new analytic solution compare well with benchmark simulations from ANSYS. So, the proposed analytic solution is a quick and easy way to approximately determine the internal load of small span suspension bridge. © 2015 Korean Society of Civil Engineers and Springer-Verlag Berlin Heidelberg
... Many authors applied this principles to study the vibrations response of suspension bridges. The authors of [19,20] used the continuous model proposed by Abdel-Ghaffar [11], and solve the system of equations by means of the multiple scale perturbative technique [21]. Recently, Airoli and Gazzola [22], trying to explain why did torsional oscillations suddenly appears before the Tacoma Narrows collapse, found out that also in isolated systems as vertical oscillations become large enough they may switch to torsional ones. ...
... Section 3 is devoted to the linearized variational system of equations projected in the modal space. The analysis of stability maps confirm the possibility of parametric resonance of 2:1 type between flexural and torsional motion already verified by other authors [11,19,20] by means of analytical procedure but on the complete non-linear original system of equations. The importance of bridge's deck sectional shape factor will be stressed out in order to explain Parametric resonance phenomenon by means of Strohual linear law for vortex-shedding excitation. ...
... The recent work by Airoli and Gazzola [22] and previous works by different authors [19,20] contain the numerical and analytical proof, respectively, that suspension bridges can suffer of internal parametric resonance. This kind of phenomenon strongly differs from the well-known ordinary resonance characteristic of linear vibrating systems. ...
