87,858 research outputs found
Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables
No full textThis paper presents space-time numerical simulation and validation of analytical predictions for the finite-amplitude forced dynamics of suspended cables. The main goal is to complement analytical and numerical solutions, accomplishing overall quantitative/qualitative comparisons of nonlinear response characteristics. By relying on an approximate, kinematically non-condensed, planar modeling, a simply supported horizontal cable subject to a primary external resonance and a 1:1, or 1:1 vs. 2:1, internal resonance is analyzed. To obtain analytical solution, a second-order multiple scales approach is applied to a complete eigenfunction-based series of nonlinear ordinary-differential equations of cable damped forced motion. Accounting for both quadratic/cubic geometric nonlinearities and multiple modal contributions, local scenarios of cable uncoupled/coupled responses and associated stability are predicted, based on chosen reduced-order models. As a cross-checking tool, numerical simulation of the associated nonlinear partial-differential equations describing the dynamics of the actual infinite-dimensional system is carried out using a finite difference technique employing a hybrid explicit-implicit integration scheme. Based on system control parameters and initial conditions, cable amplitude, displacement and tension responses are numerically assessed, thoroughly validating the analytically predicted solutions as regards the actual existence, the meaningful role and the predominating internal resonance of coexisting/competing dynamics. Some methodological aspects are noticed, along with a discussion on the kinematically approximate versus exact, as well as planar versus non-planar, cable modeling
Nonlinear longitudinal/transversal modal interactions in highly extensible suspended cables
No full textRecent 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
Nonlinear hybrid-mode resonant forced oscillations of sagged inclined cables at avoidances
No full textWe investigate non-linear forced oscillations of sagged inclined cables under planar 1:1 internal resonance at avoidance. To account for frequency avoidance phenomena and associated hybrid modes actually distinguishing inclined cables from horizontal cables, asymmetric inclined static configurations are considered. Emphasis is placed on highlighting nearly tuned 1:1 resonant interactions involving coupled hybrid modes. The inclined cable is subjected to a uniformly distributed vertical harmonic excitation at primary resonance of a high-frequency mode. Approximate non-linear partial-differential equations of motion, capturing overall displacement coupling and dynamic extensibility effect, are analytically solved based on a multi-mode discretization and a second-order multiple scales approach. Bifurcation analyses of both equilibrium and dynamic solutions are carried out via a continuation technique, highlighting the influence of system parameters on internally resonant forced dynamics of avoidance cables. Direct numerical integrations of modulation equations are also performed to validate the continuation prediction and characterize non-linear coupled dynamics in post-bifurcation states. Depending on the elasto-geometric (cable sag and inclination) and control parameters, and on assigned initial conditions, the hybrid modal interactions undergo several kinds of bifurcations and non-linear phenomena, along with meaningful transition from periodic to quasi-periodic and chaotic responses. Moreover, corresponding spatio-temporal distributions of cable non-linear dynamic displacement and tension are manifested
Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part I : theoretical formulation and model validation
No full textThis 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
Experimental and numerical studies of inclined cables: free and parametrically-forced vibrations
No full textBecause 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
11th Conference on Nonlinear Vibrations, Stability, and Dynamics of Structures, in honour of Amr Baz, Giuseppe Rega, and Fabrizio Vestroni on the occasion of their 60th birthday, Virginia Tech, USA
No full textThe effects of kinematic condensation on internally resonant forced vibrations of shallow horizontal cables
No full textThis study aims at comparing non-linear modal interactions in shallow horizontal cables with kinematically non-condensed vs. condensed modeling, under simultaneous primary external and internal resonances. Planar 1:1 or 2:1 internal resonance is considered. The governing partial-differential equations of motion of non-condensed model account for spatio-temporal modification of dynamic tension, and explicitly capture non-linear coupling of longitudinal/ vertical displacements. On the contrary, in the condensed model, a single integro-differential equation is obtained by eliminating the longitudinal inertia according to a quasi-static cable stretching assumption, which entails spatially uniform dynamic tension. This model is largely considered in the literature. Based on a multi-modal discretization and a second-order multiple scales solution accounting for higher-order quadratic effects of a infinite number of modes, coupled/uncoupled dynamic responses and the associated stability are evaluated by means of frequency- and force-response diagrams. Direct numerical integrations confirm the occurrence of amplitude-steady or -modulated responses. Non-linear dynamic configurations and tensions are also examined. Depending on internal resonance condition, system elasto-geometric and control parameters, the condensed model may lead to significant quantitative and/or qualitative discrepancies, against the non-condensed model, in the evaluation of resonant dynamic responses, bifurcations and maximal/minimal stresses. Results of even shallow cables reveal meaningful drawbacks of the kinematic condensation and allow us to detect cases where the more accurate non-condensed model has to be used
PLANAR NON-LINEAR OSCILLATIONS OF ELASTIC CABLES UNDER SUPERHARMONIC RESONANCE CONDITIONS
No full textPlanar non-linear oscillations of elastic cables under order two and three superharmonic resonance conditions are studied. As referred to one ordinary equation of motion, second order perturbation analyses are developed and the solutions are used to enlight the features of the two dynamic phenomena for technical cables with various sag-to-span ratios. Some aspects of interaction between the two main superharmonic components occurring in the motion are discussed, and the results of numerical integrations of the original equation are presente
Numerical simulations of chaotic dynamics in a model of elastic cable
No full textThe finite motions of a suspended elastic cable subjected to a planar harmonic excitation can be studied accurately enough through a single ordinary-differential equation with quadratic and cubic nonlinearities. The possible onset of chaotic motion for the cable in the region between the one-half subharmonic resonance condition and the primary one is analysed via numerical simulations. Chaotic charts in the parameter space of the excitation are obtained and the transition from periodic to chaotic regimes is analysed in detail by using phase-plane portraits, Poincaré maps, frequency-power spectra, Lyapunov exponents and fractal dimensions as chaotic measures. Period-doubling, sudden changes and intermittency bifurcations are observed
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