Nonlinear oscillations of cables under harmonic loading using analytical and finite element models (original) (raw)
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Nonlinear Dynamics, 2007
Two-to-one resonant multi-modal dynamics of horizontal/inclined cables. Part II: internal resonance activation, reduced-order models and nonlinear normal modes. Nonlinear Dynamics, 48 (3). pp. 253-274. ISSN 1573-269X http://strathprints.strath.ac.uk/18556/ This is an author produced version of a paper published in Nonlinear Dynamics, 48 (3). pp. 253-274. ISSN 1573-269X. This version has been peer-reviewed but does not include the final publisher proof corrections, published layout or pagination.
Internal resonance and non-linear response of a cable under periodic excitation
Journal of Sound and Vibration, 1991
The coupled non-linear equations of motion of a sagged cable in its first symmetric mode of in-plane and out-of-plane oscillations are solved by the method of multiple scales for its forced vibration response. The cases of both internal and external resonances are considered. A uniform lateral load is assumed to act along with an in-plane harmonic component, similar to a situation under vortex-induced oscillation. The quadratic nonlinearity terms in the equations of motion are shown to affect the cable behaviour significantly when the in-plane frequency is about twice the out-of-plane frequency of oscillation. A stability analysis is performed on the steady state solutions. The effect of cable sag on these solutionns and their stability is studied indirectly through the internal resonance parameter. The influence of the lateral load component, with and without internal resonance, on the stability regions is discussed.
Dynamics of cable structures – Modeling and applications
2013
v The objective of the present work is to re-examine and appropriately modify the geometrically exact beam theory, originally developed by Simo, and develop a nonlinear finite-element formulation to describe the static and dynamic behavior of flexible electrical equipment cables. The work is motivated by the need to better understand and predict the highly nonlinear response of flexible electrical conductors to earthquake excitations. Dynamic interaction between flexible cables and interconnected substation equipment is in fact believed to explain some of the severe damage sustained by such equipment in recent earthquakes. In the first part of this report, the nonlinear equations of motion of a beam undergoing large displacements and rotations are derived from the 3D theory of continuum mechanics by use of the virtual power equation. A linear viscoelastic constitutive equation and an additional mass proportional damping mechanism are used to account for energy dissipation. The weak ...
Nonlinear Dynamics, 2006
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.
Catenary-induced geometric nonlinearity effects on cable linear vibrations
Journal of Sound and Vibration, 2018
This paper investigates the free undamped vibrations of cables of arbitrary sag and inclination according to the catenary theory. The proposed approach accounts for the catenary effect on the static profile around which the cable motion is defined. Considering first order geometric nonlinearities, exact expression of the curvature is obtained along with the ensuing correction of the well known Irvine parameter. Taking into account the new characterization, different regions of shallow and non-shallow profiles are identified for various inclinations. In view of such classification, the analysis carried out on cable linear modal properties shows the emergence of new dynamic features such as additional hybrid modes and internal resonances. Analytical and numerical results reduce to those obtained by classic formulations in the cases of both horizontal and inclined shallow/non-shallow cables.
Nonlinear hybrid-mode resonant forced oscillations of sagged inclined cables at avoidances
Journal of Computational and Nonlinear Dynamics, 2007
We investigate nonlinear 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 nonlinear partial-differential equations of motion, capturing overall displacement coupling and dynamic extensibility effect, are analytically solved based on a multimode discretization and a second-order multiple scale 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 nonlinear 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 nonlinear phenomena, along with meaningful transition from periodic to quasiperiodic and chaotic responses. Moreover, corresponding spatio-temporal distributions of cable nonlinear dynamic displacement and tension are manifested. to 151.100.30.40. Redistribution subject to ASME license or copyright; see http://www.asme.org/terms/Terms\_Use.cfm Fig. 1 "a… Schematic model of a sagged inclined cable. "b… Planar frequency spectrum and avoidance phenomena of inclined cable with = 30 deg.
Non-Linear Dynamics of Electrical Equipment Cables
Proceedings of the 4th International Conference on Computational Methods in Structural Dynamics and Earthquake Engineering (COMPDYN 2013)
In power substations, electrical cables are commonly observed in configurations that cannot be explained by tension stress states only. Moreover, during extreme excitations such as earthquakes, they can be subjected to large displacements and rotations, as well as forces that can lead to failure of both the cables and the equipment to which they are connected. In this paper a three-dimensional geometrically exact beam model is presented and used to describe the static and dynamic behavior of electrical equipment cables. The model is implemented in MATLAB and applied to a flexible conductor recently tested at the SEESL laboratory at the University at Buffalo. Both static and dynamic numerical applications are performed. First the initial configuration of the tests is obtained by imposing the end displacements and rotations to a straight and unstrained cable. Then the cable is subjected to resonant harmonic out-of-plane and in-plane horizontal ground motions. Due to the high frequency components resulting from the axial response of the cable, and to the fact that the Newmark time-integration algorithm used herein generally fails to conserve energy, very small time steps are needed for the algorithm to converge. Although algorithms have been proposed in the literature, which introduce numerical damping to stabilize the computations, the introduction of real forms of damping in the model and the simulation of the dynamic experimental tests performed at the University at Buffalo are object of current work.
Revisited modelling and multimodal nonlinear oscillations of a sagged cable under support motion
Meccanica, 2016
Sagged cable vibrations caused by support motion and possible external loading are investigated via the four-degree-of-freedom model proposed in Benedettini et al. (J Sound Vib 182(5):775-798, 1995). The model has a considerable potential in terms of forcing cases to be possibly addressed, with the physical motion of the supports naturally giving rise to a variety of external and parametric excitation terms. Dynamics of the system is studied close to the multiple internal resonance at cable crossover, which involves two in-plane and two out-of plane vibration modes. Solutions are found by the multiple time scale method. In the numerical investigation, attention is focused on the effects of planar support motion (symmetric and/or antisymmetric) at primary resonance, with the addition of planar symmetric external excitation entailing a nice cancellation phenomenon in the system response. Results are discussed also in the background of theoretical and experimental outcomes available in the literature. Comparison with a computer simulation of original equations of motion shows that analytical results are correct for moderately large oscillations, whereas a different scenario of multimodal responses may occur at higher excitation amplitudes. The nonlinear modal coupling is investigated through bifurcation scenarios and other dynamics tools, showing also transitions to complex response regimes. Keywords Suspended cable Á Support motion Á External/parametric excitations Á Nonlinear oscillations Á Multimodal response Dedicated to the memory of Francesco Benedettini, who was the first assistant and a lifelong friend of GR, as well as the unforgettable first mentor of DZ.
Journal of Sound and Vibration, 2008
Srinil, N. and Rega, G. (2008) Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables. Journal of Sound and Vibration, 315 (3). pp. 394-413. ISSN 0022-460X Strathprints is designed to allow users to access the research output of the University of Strathclyde. Srinil, N. and Rega, G. (2008) Space-time numerical simulation and validation of analytical predictions for nonlinear forced dynamics of suspended cables. Journal of Sound and Vibration, 315 (3). pp. 394-413. ISSN 0022-460X http://strathprints.strath.ac.uk/18550/ This is an author produced version of a paper published in Journal of Sound and Vibration, 315 (3). pp. 394-413. ISSN 0022-460X. This version has been peer-reviewed but does not include the final publisher proof corrections, published layout or pagination.
On large-amplitude vibrations of cables
Journal of Sound and Vibration, 1987
A paper by AI-Noury and Ali entitled "Large-amplitude vibrations of parabolic cables" was recently published in this Journal . It appeared to us that their aim was to refocus on the main aspects of the non-linear dynamics of cables which have been the subject of several recent works and to re-examine the procedures most frequently used to handle the spatial and temporal dependence. Although in reference [1] forced coupled in-plane and out-of-plane oscillations were studied, a problem not previously analyzed, the authors made no mention of a certain number of earlier contributions [2-7] which might have helped in their study of this problem.