Whirling motion of a shallow cable with viscous dampers

The paper deals with the non-linear dynamic analysis of cables with a pair of viscous dampers close to one support. Such cables are characterized by a sag-to-chord-length ratio below 0.02, for which natural frequencies for the vertical and the horizontal vibrations are pair-wise close. Under resonance the non-linear coupling of pairs of modes may cause whirling harmonic motions around the chord line. Whirling motion may occur after bifurcation from single-mode response for harmonic loads in either vertical or horizontal direction. The non-linear features are included in the two coupled modes, while all other modes are treated as linear. The motion is discretized by expansion in terms of the damped complex eigenfunctions. The applied base functions fulfil the transition condition at the damper, leading to fast convergence of the expansion. It is demonstrated that the behaviour of the whirling motion is controlled primarily by the damper acting in the direction of the unloaded mode, whereas the magnitude of the damper in the loaded mode is less important. If the dampers in the vertical and horizontal direction are close to the optimal value of the corresponding taut cable case, substantial reduction of the vibration level of the whirling mode as well as the frequency interval of its occurrence is attained.     

Transient vibration phenomena in deep mine hoisting cables. Part 2: Numerical simulation of the dynamic response

A simulation model is presented which investigates the dynamic response of a deep mine hoisting cable system during a winding cycle. The response, namely the lateral motions of the catenary cable and the longitudinal motion of the vertical rope with conveyance is observed on the fast time scale, and the slow time scale is introduced to monitor the variation of slowly varying parameters of the system. The cable equivalent proportional damping parameters, and periodic excitation functions resulting from the cross-over cable motion on the winder drum are identified. Subsequently, the model is solved numerically using parameters of a double-drum multi-rope system. Since the system eigenvalues are widely spread and the problem is of stiff nature, the numerical simulation is conducted using a stiff solver. The results of the simulation demonstrate various transient non-linear resonance phenomena arising in the system during the wind. The nominal ascending cycle simulation results reveal adverse dynamic behaviour of the catenary largely due to the autoparametric interactions between the in- and out-of-plane modes. Principal parametric resonances of the lateral modes also occur, and conditions for autoparametric interactions between the lateral and longitudinal modes arise. Additionally, a transition through a number of primary longitudinal resonances takes place during the wind. The adverse dynamic motions in the system promote large oscillations in the cable tension which must be considered significant with respect to fatigue of the cable. It is noted that a small change in the winding velocity may cause large changes in the dynamic response due to the resonance region shifts. Consequently, the resonance modal interactions can be avoided, to a large extent, if the winding velocity is increased to an appropriate level.     

Transient vibration phenomena in deep mine hoisting cables. Part 1: Mathematical model

The classical moving co-ordinate frame approach and Hamilton's principle are employed to derive a distributed-parameter mathematical model to investigate the dynamic behaviour of deep mine hoisting cables. This model describes the coupled lateral-longitudinal dynamic response of the cables in terms of non-linear partial differential equations that accommodate the non-stationary nature of the system. Subsequently, the Rayleigh-Ritz procedure is applied to formulate a discrete mathematical model. Consequently, a system of non-linear non-stationary coupled second order ordinary differential equations arises to govern the temporal behaviour of the cable system. This discrete model with quadratic and cubic non-linear terms describes the modal interactions between lateral oscillations of the catenary cable and longitudinal oscillations of the vertical rope. It is shown that the response of the catenary-vertical rope system may feature a number of resonance phenomena, including external, parametric and autoparametric resonances. The parameters of a typical deep mine winder are used to identify the depth locations of the resonance regions during the ascending cycles with various winding velocities.     

Effects of cables damage on vertical and torsional eigenproperties of suspension bridges

This paper presents a continuum model for the nonlinear coupled vertical and torsional vibrations of suspension bridges with arbitrary damage in one main cable and, after pursuing a suitable linearization of the equations of motion, an investigation of damage effects on modal parameters. Damage is modeled as a diffused loss of cross-section representing the typical effect of fretting fatigue and it is introduced in the formulation by enforcing relevant literature results providing analytical solution for the static response of damaged suspended cables. The coupled nonlinear equations of motion of the damaged bridge, including the effects of shear deformation, rotary inertia and warping of the cross-section of the girder, are derived by application of Hamilton?s principle. In this way, the equations of motion available in the literature for undamaged suspension bridges are generalized to the presence of arbitrary damage in one main cable and the resulting eigenfrequencies and eigenfunctions are derived in an analytical fashion. An extensive parametric investigation is finally presented to discuss damage effects on eigenfunctions and eigenfrequencies under variation of practically meaningful parameters.     

Experimental study of the nonlinear dynamic characteristics of suspended taut steel cables using a 3-D motion analysis system

This paper presents an experimental study of the nonlinear dynamic characteristics of taut steel cables using a 3-D motion analysis system. In the experiment, the taut cables have one end fixed and the other end subject to harmonic vertical excitation. The 3-D motion analysis system can simultaneously record (with high resolution) the instant 3-D coordinates of the multiple markers fixed on a vibrating cable; this distinguishes it from other experimental systems used in vibration studies, in which the vibration of only one single point can be recorded during each individual testing. With the 3-D motion analysis system, this experimental study presents a distinctive interpretation of the dynamic characteristics of taut cables in spatial domain (based on the mode-shape information of the entire cable), in addition to one in time domain (based on real-time traces of one single point). This paper introduces the 3-D motion analysis system and experimental setup, discusses practical experimental procedures, and presents a detailed analysis of three sets of experimental vibration data of three taut steel cables with different small sags. The frequency response curves were obtained for three cables. For one of the three taut cables, more informative vibration data were recorded; this cable was studied in greater detail via modal analysis using a modal decomposition technique and nonlinear time-series analysis.     

Parametric-resonance-induced cable vibrations in network cable-stayed bridges. A continuum approach

There is a wealth of evidence to suggest that the bearing cables of cable-stayed bridges may experience large-amplitude oscillations, attributed in general to parametric resonance with the girder vibrations. A common coutermeasure consists of connecting the principal stays together with secondary cables to form a network and, here, optimal cable arrangments will be discussed when such a network is uniform and triangular meshed. The present approach is qualitative, and basically consists of homogenizing the cable net to an orthotropic elastic membrane, and then considering an auxiliary structure where the bridge girder, instead of being supported by the cable network, is supported by wedge-shaped membranes. The elastic solution under uniformly distributed loads, found using Lekhnitskii's approach, is the starting point for the discussion of the system in dynamic equilibrium. Having established a correspondence between the cable-net size and shape and the elastic moduli of the homogenized membrane, simple formulas are obtained to describe the global bridge vibration, as well as the local oscillations of the cables. It is then possible to estimate the girder and cable-net characteristic frequencies, to evaluate those conditions possibly leading to parametric resonance and, with respect to these variables, to determine optimal cable arrangements. This method is finally applied to the paradigmatic example of the Normandy Bridge.     

FREE NON-LINEAR VIBRATION OF A ROTATING THIN RING WITH THE IN-PLANE AND OUT-OF-PLANE MOTIONS

Free non-linear vibration of a rotating thin ring with a constant speed is analyzed when the ring has both the in-plane and out-of-plane motions. The geometric non-linearity of displacements is considered by adopting the Lagrange strain theory for the circumferential strain instead of the infinitesimal strain theory. By using Hamilton's principle, the coupled non-linear partial differential equations are derived, which describe the out-of-plane bending and torsional motions as well as the in-plane bending and extensional motions. During deriving the equations of motion, we discuss how to model the circumferential stress and strain in order to consider the geometric non-linearity. Four models are established: three non-linear models and one linear model. For the four models, the linearized equations of motion are obtained in the neighbourhood of the steady state equilibrium position. Based on the linearized equations of the four cases, the natural frequencies are computed at various rotational speeds and then they are compared. Through the comparison, this study recommends which model is appropriate to describe the non-linear behaviour more precisely.     

A parametric multi-body section model for modal interactions of cable-supported bridges

A parametric section model is formulated to synthetically describe the geometrically nonlinear dynamics of cable-stayed and suspended bridges through a planar elastic multi-body system. The four-degrees-of-freedom model accounts for both the flexo-torsional motion of the bridge deck and for the transversal motion of a pair of hangers or stay cables. After linearization around the pre-stressed static equilibrium configuration, the coupled equations of motion governing the global deck dynamics and the local cable motion are obtained. A multi-parameter perturbation method is employed to solve the modal problem of internally resonant systems. The perturbation-based modal solution furnishes, first, explicit formulae for the parameter combinations which realize the internal resonance conditions and, second, asymptotic approximations of the resonant frequencies and modes. Attention is focused on the triple internal resonance among a global torsional mode of the deck and two local modes of the cables, due to the relevant geometric coupling which maximizes the modal interaction. The asymptotic approximation of the modal solution is found to finely describe the multiple veering phenomenon which involves the three frequency loci under small variation of the most significant mechanical parameters, including terms of structural coupling or disorder. Moreover, the veering amplitude between any two of the three frequency loci can be expressed as an explicit parametric function. Finally, the disorder is recognized as the only parameter governing a complex phenomenon of triple modal hybridization involving all the resonant modes. The entire hybridization process is successfully described by an energy-based localization factor, presented in a new perturbation-based form, valid for internally resonant system.     

Three-dimensional non-linear coupling and dynamic tension in the large-amplitude free vibrations of arbitrarily sagged cables

This paper presents a model formulation capable of analyzing large-amplitude free vibrations of a suspended cable in three dimensions. The virtual work-energy functional is used to obtain the non-linear equations of three-dimensional motion. The formulation is not restricted to cables having small sag-to-span ratios, and is conveniently applied for the case of a specified end tension. The axial extensibility effect is also included in order to obtain accurate results. Based on a multi-degree-of-freedom model, numerical procedures are implemented to solve both spatial and temporal problems. Various numerical examples of arbitrarily sagged cables with large-amplitude initial conditions are carried out to highlight some outstanding features of cable non-linear dynamics by accounting also for internal resonance phenomena. Non-linear coupling between three- and two-dimensional motions, and non-linear cable tension responses are analyzed. For specific cables, modal transition phenomena taking place during in-plane vibrations and ensuing from occurrence of a dominant internal resonance are observed. When only a single mode is initiated, a higher or lower mode can be accommodated into the responses, making cable spatial shapes hybrid in some time intervals.     

Equations of motion for damped longitudinal,torsional and flexural oscillations of piezoelectric bars

The equations of motion for pure (mutually independent) longitudinal, torsional and flexural oscillations of very thin piezoelectric bars are derived systematically and in detail first in integral and then in differential form. During derivation attention is paid to the effect of external and internal damping and the method of exciting the individual oscillations is considered.     

Non-linear vibrations of cables considering loosening

A cable cannot resist the axial compressive force that may be induced during large amplitude vibrations. In this paper, the effect of cable loosening on non-linear vibrations of flat-sag cables is discussed by using the finite difference method that can express cable loosening. In the present method, flexural rigidity and damping of the cable are considered in the equations of motion of a cable in order to handle the numerical instability. The effect of cable loosening is evaluated explicitly in the present paper. Furthermore, non-linear vibration properties are evaluated for various parameters under periodic and step vertical loading. The effect of cable loosening on response under vertical periodic time-varying load is small and it is possible for the sag-to-span ratio to roughly equal the ratio for modal transition. The loosening under the vertical step loading in the direction opposite to the gravity appears at almost the same sag-to-span ratio.     

Modeling and analysis of the in-plane vibration of a complex cable-stayed bridge

The in-plane vibration of a complex cable-stayed bridge that consists of a simply-supported four-cable-stayed deck beam and two rigid towers is studied. The nonlinear and linear partial differential equations that govern transverse and longitudinal vibrations of the cables and transverse vibrations of segments of the deck beam, respectively, are derived, along with their boundary and matching conditions. The undamped natural frequencies and mode shapes of the linearized model of the cable-stayed bridge are determined, and orthogonality relations of the mode shapes are established. Numerical analysis of the natural frequencies and mode shapes of the cable-stayed bridge is conducted for various symmetrical and non-symmetrical bridge cases with regards to the sizes of the components of the bridge and the initial sags of the cables. The results show that there are very close natural frequencies when the bridge model is symmetrical and/or partially symmetrical, and the mode shapes tend to be more localized when the bridge model is less symmetrical. The relationships between the natural frequencies and mode shapes of the cable-stayed bridge and those of a single fixed–fixed cable and the single simply-supported deck beam are analyzed. The results, which are validated by commercial finite element software, demonstrate some complex classical resonance behavior of the cable-stayed bridge.     

Optimal design of passive viscous dampers for controlling the resonant response of orthotropic plates under high-speed moving loads

In this article, the resonant response of plates traversed by moving loads is addressed being its main application the dynamic performance of railway bridges under high-speed traffic. An innovative alternative to reduce the deck inadmissible oscillations that may appear in short simply supported structures in resonant conditions is proposed, based on artificially increasing the superstructure damping by retrofitting the deck with fluid viscous dampers. A particular auxiliary structure transforming the deck vertical deflection into relative movement within the devices is envisaged, being the main objectives of the study to optimise the retrofitting system parameters and to prove its efficiency under the action of railway vehicles. For these purposes, the retrofitted deck behaviour is first investigated using an orthotropic plate model under harmonic excitation. On the basis of an analytical approach, a dimensionless version of the equations of motion is presented, the governing parameters are extracted and an intensive sensitivity analysis of the plate response is performed. Finally, analytical closed-form expressions for the optimal dampers constants are derived and their adequacy is numerically evaluated. To this end, an existing bridge belonging to the Spanish Railway network is analysed using a three-dimensional finite element code specifically programmed by the authors for this application. In the end the controlling effect of the retrofitting system and the applicability of the optimal parameters analytical expressions are proven for a wide range of circulating velocities.     

Fiber optic aerial cables

While conventional cable elements can tolerate strains on the order of 1%, optical fibers cannot. As aerial cables may suffer contraction and elongation from low temperatures and high ice or wind loads, or high temperatures, respectively, proper designs have to provide the cables with length margins within which the fibers are protected against mechanical loads, except for a very limited bending stress. The well-suited loose tube buffer design, the way of calculation, and a list of reference cable plants are presented.     

Fiber optic aerial cables

While conventional cable elements can tolerate strains on the order of 1%, optical fibers cannot. As aerial cables may suffer contraction and elongation from low temperatures and high ice or wind loads, or high temperatures, respectively, proper designs have to provide the cables with length margins within which the fibers are protected against mechanical loads, except for a very limited bending stress. The well-suited loose tube buffer design, the way of calculation, and a list of reference cable plants are presented.     

Response characteristics of local vibrations in stay cables on an existing cable-stayed bridge

This paper examines local parametric vibrations in the stay cables of a cable-stayed bridge. The natural frequencies of the global modes are obtained by using a three-dimensional FE model. The global motions generated by (1) sinusoidal excitations using exciter, (2) a traffic loading, and (3) an earthquake are analyzed by using the modal analysis method or the direct integration method. The local vibration of stay cable is calculated by using a model in which inclined cable is subjected to time-varying displacement at one support during global motions. This paper describes the properties of the local vibrations in stay cables under these dynamic loadings by using an existing cable-stayed bridge.     

The effect of cable loosening on seismic response of a prestressed concrete cable-stayed bridge

In conventional non-linear seismic analyses of cable-stayed bridges, the non-linear characteristics of the girders, stay cables and towers are considered. The non-linearity caused by cable loosening should also be considered because a large axial force fluctuation is generated in the cables of a prestressed concrete (PC) cable-stayed bridge that is subjected to strong seismic motion. In this paper, the possibility of the cable loosening in a PC cable-stayed bridge is discussed by using a cable model that can express the cable loosening. Furthermore, the effect of the cable loosening on the responses of the cables, girder and towers is evaluated using the mean value for three seismic waves. Numerical analytic results imply that the cable loosening appears in the bottom cables of the multi-cable system and the dynamic response of the bridge is slightly increased.     

Longitudinal,transverse, and torsional vibrations and stabilities of axially moving single-walled carbon nanotubes

Thanks to the brilliant mechanical properties of single-walled carbon nanotubes (SWCNTs), they are suggested as high speed nanoscale vehicles. To date, various aspects of vibrations of SWCNTs have been addressed; however, vibrations and instabilities of moving SWCNTs have not been thoroughly assessed. Herein, vibrational properties of an axially moving SWCNT with simply supported ends are studied using nonlocal Rayleigh beam theory. Employing assumed mode and Galerkin methods, the discrete governing equations pertinent to longitudinal, transverse, and torsional motions of the moving SWCNT are obtained. The resulting eigenvalue equations are then numerically solved. The speeds corresponding to the initiation of the instability within the moving nanostructure are calculated. The roles of the speed of the moving SWCNT, small-scale parameter, and aspect ratio on the characteristics of longitudinal, transverse, and torsional vibrations of axially moving SWCNTs are scrutinized. The obtained results show that the appearance of the small-scale parameter would result in the occurrence of both divergence and flutter instabilities at lower levels of the speed.     

Aeroelastic analysis and nonlinear dynamics of an elastically mounted wing

The response of an elastically mounted wing that is free to plunge and pitch, supported by nonlinear translational and torsional springs, and interacting with an incoming stream is analyzed. A tightly coupled model of the wing flow interaction is developed. A three-dimensional code based on the unsteady vortex lattice method is used for the prediction of the unsteady aerodynamic loads. The response of the wing shows a sequence of static and dynamic bifurcations and chaotic motions when increasing the flow speed. Pairs of stable solutions are observed over the different response regimes. The effects of the gust and structural nonlinearity on the wing's response are also investigated. The results show that gust may lead to jumps between the pairs of solutions for static and dynamic equilibrium responses without impacting the boundaries of the different response regimes. As for the effect of the structural nonlinearity, increasing the nonlinear coefficient of the stiffness of the torsional spring yields lower static deflections and amplitudes of the limit cycle oscillations.