Nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables

Recent research literature mostly deals with nonlinear resonant dynamics of low-extensible cables involving transversal modes. Herein, we aim to investigate geometrically nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables, whose material properties are assumed to be linearly elastic. Depending on cable elasto-geometric properties, the spectrum of low-order planar frequencies manifests primary and secondary frequency crossover phenomena of transversal/transversal and longitudinal/transversal modes, respectively. By focusing on 1:1 internal resonances, nonlinear equations of finite-amplitude, harmonically forced and damped, cable motion are considered, fully accounting for overall inertia and displacement coupling effects. Meaningful quadratic nonlinear contributions of non-resonant, higher-order, longitudinal modes are highlighted via a multimode-based, second-order multiple scales solution. Overall coupled/uncoupled dynamic responses, bifurcations, stability and space-time-varying displacements due to longitudinal/transversal (vs. transversal/transversal) modal interactions at secondary (vs. primary) crossovers are analytically and numerically evaluated, along with the resonant longitudinal mode-induced dynamic forces

Experimental and numerical studies of inclined cables: free and parametrically-forced vibrations

Because of few experimental studies in the inclined cable literature, this paper is aimed at experimental modelling and investigating the linear free and nonlinear forced vibrations of sagged inclined cables, by discussing the relevant outcomes in the background of theoretical and numerical achievements. Attention is paid to the identification of cable hybrid modes due to system asymmetry, which gives rise to an avoidance phenomenon in the natural frequency spectrum, and to the investigation of some typical 3-D nonlinear dynamics involving the simultaneous parametric/external excitation due to a harmonically time-varying support movement. Large-amplitude out-of-plane/in-plane multi-modal interactions due to non-planar/planar internal resonances are experimentally observed and complemented by space-time numerical simulation of the associated, geometrically nonlinear, partial-differential equations of parametrically-forced cable motion. Overall, the experimental and numerical results highlight the fundamental linear/nonlinear dynamic characteristics of inclined cables, and the crucial role played by the asymmetry induced by cable inclination, in addition to the significant effects of cable sag and dynamic extensibility

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II : internal resonance activation, reduced-order models and nonlinear normal modes

Resonant multi-modal dynamics due to planar 2:1 internal resonances in the nonlinear, finite-amplitude, free vibrations of horizontal/inclined cables are parametrically investigated based on the second-order multiple scales solution in Part I [1]. The already validated kinematically non-condensed cable model accounts for the effects of both non-linear dynamic extensibility and system asymmetry due to inclined sagged configurations. Actual activation of 2:1 resonances is discussed, enlightening on a remarkable qualitative difference of horizontal/inclined cables as regards non-linear orthogonality properties of normal modes. Based on the analysis of modal contribution and solution convergence of various resonant cables, hints are obtained on proper reduced-order model selections from the asymptotic solution accounting for higher-order effects of quadratic nonlinearities. The dependence of resonant dynamics on coupled vibration amplitudes, and the significant effects of cable sag, inclination and extensibility on system non-linear behavior are highlighted, along with meaningful contributions of longitudinal dynamics. The spatio-temporal variation of non-linear dynamic configurations and dynamic tensions associated with 2:1 resonant non-linear normal modes is illustrated. Overall, the analytical predictions are validated by finite difference-based numerical investigations of the original partial-differential equations of motion

New model for vortex-induced vibration of catenary riser

This paper presents a new theoretical model capable of predicting the vortex-induced vibration response of a steel catenary riser subject to a steady uniform current. The equations governing riser in-plane/out-ofplane (cross-flow/in-line) motion are based on a pinned beam-cable model accounting for overall effects of bending, extensibility, sag, inclination and structural nonlinearities. The empirically hydrodynamic model is based on nonlinear wake oscillators describing the fluctuating lift/drag forces. Depending on the potentially vortex-induced modes and system parameters, a reduced-order fluid-structure interaction model is derived which entails a significantly reduced computational time effort. Parametric results reveal maximum response amplitudes of risers, along with the occurrence of uni-modal lock-in phenomenon

Multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures with geometric nonlinearities

A general low-order fluid-structure interaction model capable of evaluating the multi-mode interactions in vortex-induced vibrations of flexible curved/straight structures is presented. Cross-flow motions due to unsteady lift forces of inclined sagged cables and tensioned beams in uniform currents are investigated. In contrast to a linear equation governing the transverse motion of straight beams or cables typically considered in the literature, coupled horizontal/vertical (axial/transverse) displacements and geometric nonlinearities of curved cable (straight beam) are accounted for. A distributed nonlinear wake oscillator is considered in the approximation of space-time varying hydrodynamics. This semi-empirical fluid force model in general depends on the mass-damping parameter and has further been modified to capture both the effects of varying initial curvatures of the inclined cylinder and the Reynolds number. Numerical simulations are performed in the case of varying flow velocities and parametric results highlight several meaningful aspects of vortex-induced vibrations of long flexible cylinders. These comprise multi-mode lock-in, sharing, switching and interaction features in the space and time domains, the estimated maximum modal and total amplitudes, the resonant nonlinear modes of flexible cylinders and their space-time modifications, and the influence of fluid/structure parameters. A shortcoming of single-mode or linear structural model is underlined. Some quantitative and qualitative comparisons of numerical/experimental results are discussed to demonstrate the validity and required improvement of the proposed modelling and analysis predictions

Nonlinear multi-mode interactions in subsea risers undergoing vortex-induced vibrations

This paper investigates nonlinear multi-mode interactions in subsea risers undergoing vortex-induced vibrations based on a computationally efficient reduced-order fluid-structure interaction model. Cross-flow responses as a result of a steady uniform current are considered. The geometrically nonlinear equations of riser motion are coupled with nonlinear wake oscillators which have been modified to capture the effect of initial curvatures of curved cylinder and to approximate the space-time varying hydrodynamic lift forces. The main objectives are to provide new insights into the vortex-induced vibration characteristics of risers under external and internal resonances and to distinguish nonlinear dynamic behaviors between curved catenary and straight toptensioned risers. The analyses of multi-mode contributions, lock-in regimes, response amplitudes, resonant nonlinear modes and curvatures are carried out and several interesting aspects are highlighted

Vortex-induced vibration of catenary riser: reduced-order modeling and lock-in analysis using wake oscillator

A new reduced-order model capable of analyzing the vortex-induced vibration of catenary riser in the ocean current has been developed. This semi analytical-numerical approach is versatile and allows for a significant reduction in computational effort for the analysis of fluid-riser interactions. The incoming current flow is assumed to be steady, uniform, unidirectional and perpendicular to the riser plane of initial equilibrium curvatures

Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I : theoretical formulation and model validation

This paper is first of the two papers dealingwith analytical investigation of resonant multimodal dynamics due to 2:1 internal resonances in the finite-amplitude free vibrations of horizontal/inclined cables. Part I deals with theoretical formulation and validation of the general cable model. Approximate nonlinear partial differential equations of 3-D coupled motion of small sagged cables - which account for both spatio-temporal variation of nonlinear dynamic tension and system asymmetry due to inclined sagged configurations - are presented. A multidimensional Galerkin expansion of the solution ofnonplanar/planar motion is performed, yielding a complete set of system quadratic/cubic coefficients. With the aim of parametrically studying the behavior of horizontal/inclined cables in Part II [25], a second-order asymptotic analysis under planar 2:1 resonance is accomplished by the method of multiple scales. On accounting for higher-order effectsof quadratic/cubic nonlinearities, approximate closed form solutions of nonlinear amplitudes, frequencies and dynamic configurations of resonant nonlinear normal modes reveal the dependence of cable response on resonant/nonresonant modal contributions. Depending on simplifying kinematic modeling and assigned system parameters, approximate horizontal/inclined cable models are thoroughly validated by numerically evaluating statics and non-planar/planar linear/non-linear dynamics against those of the exact model. Moreover, the modal coupling role and contribution of system longitudinal dynamics are discussed for horizontal cables, showing some meaningful effects due to kinematic condensation