F. Benedettini

Non-linear dynamics of an elastic cable under planar excitation

Abstract The phenomena of the finite forced dynamics of a suspended cable associated with the quadratic and cubic non-linearities in the equations of motion are studied. A high-order perturbation analysis for the primary resonance is accomplished and numerical results are presented for the frequency-response equation and the region of instability of the steady-state solutions. Multivaluedness of the response curves is shown to occur with different characteristics depending on the cable and forcing parameters. The dependence of the response on the initial conditions is examined by means of the trajectories of the unsteady-state motions.

Experimental Investigation of the Nonlinear Response of a Hanging Cable. Part II: Global Analysis

An experimental model of an elastic cable carrying eight concentrated masses and hanging at in-phase or out-of-phase vertically moving supports is considered. The system parameters are adjusted to approximately realize multiple 1:1 and 2:1 internal resonance conditions involving planar and nonplanar, symmetric and antisymmetric modes. Response measurements are made in various frequency ranges including meaningful external resonance conditions. A ‘local’ analysis of the system response is made on the basis of numerous amplitude-frequency and amplitude-forcing plots obtained in different ranges of the control parameter space. Attention is mainly devoted to the detection of the main features of the regular motions exhibited by the system, and to the analysis of the relevant phenomena of nonlinear modal interaction, competition, and local bifurcation between planar and nonplanar regular responses. The resulting picture appears very rich and varied.

Planar non-linear oscillations of elastic cables under superharmonic resonance conditions

Abstract Planar 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 presented.

Periodic and chaotic motions of an unsymmetrical oscillator in nonlinear structural dynamics

Abstract The chaotic dynamics of a harmonically-excited single-degree-of-freedom unsymmetric system in continuum nonlinear structural dynamics is studied in some detail through computer simulations. The quadratic and cubic nonlinearities occurring in the motion equation model the geometrical and mechanical characteristics of a suspended elastic cable vibrating in its plane and actually exhibiting a single equilibrium position. Regions of different periodic or chaotic responses in the forcing control parameter space are obtained. Three main frequency ranges are distinguished, two located in the neighbourhood of the order one-half and one-third subharmonic resonances of the system, the third one covering the zones of order three and two superharmonic resonances. Several types of bifurcations and strange attractors are observed and identified through various dynamic measures, both qualitative and quantitative. The algorithmic experience about their use in situations with different “chaoticity” is augmented.

Prediction of bifurcations and chaos for an asymmetric elastic oscillator

Abstract The nonlinear chaotic response of a harmonically forced elastic oscillator with quadratic and cubic nonlinearities is studied. The possibility of obtaining reliable prediction of bifurcations and chaos through combined use of stability analysis of low-order approximate solutions and accurate localized point-by-point and cell mapping computer simulations is examined. Some satisfactory results are obtained for the bifurcation predictive capability and the location of regions where chaos actually occurs.

Numerical and geometrical analysis of bifurcation and chaos for an asymmetric elastic nonlinear oscillator

An asymmetric nonlinear oscillator representative of the finite forced dynamics of a structural system with initial curvature is used as a model system to show how the combined use of numerical and geometrical analysis allows deep insight into bifurcation phenomena and chaotic behaviour in the light of the system global dynamics.Numerical techniques are used to calculate fixed points of the response and bifurcation diagrams, to identify chaotic attractors, and to obtain basins of attraction of coexisting solutions. Geometrical analysis in control-phase portraits of the invariant manifolds of the direct and inverse saddles corresponding to unstable periodic motions is performed systematically in order to understand the global attractor structure and the attractor and basin bifurcations.

Periodic Solutions Leading to Chaos in an Oscillator with Quadratic and Cubic Nonlinearities

Regions of periodic and chaotic response, types of bifurcation and strange attractors of an unsymmetric oscillator of interest in structural dynamics are analyzed. The bifurcation predictive capability of the stability analysis of simple approximate solutions is discussed.

Nonregular Regimes of Monodimensional Mechanical Systems with Initial Curvature: Experiments and Time Series Analysis

The 3D finite dynamics of monodimensional elastic structures with initial curvature is particularly rich and varied because of the presence of both even and odd nonlinearities, the former being directly linked to the initial curvature [1]. This work is concerned with the analysis of experimental models of two different systems belonging to that class. The first is a discrete model of an elastic suspended cable excited by vertical, sinusoidally varying, motion of the hanging points. The second is a steel model of a double hinged circular arch excited by a vertical, sinusoidally varying, concentrated force at its tip. In strongly developed nonlinear regimes, interesting phenomena linked to the nonlinear modal interaction appear in such systems over a wide range of excitation frequencies, owing to the very close sequence of primary and secondary resonance conditions. Moreover, changing the sag to span ratio of the cable or adding a vertical dead load on the tip of the arch it is possible to obtain various internal resonance conditions which further exhalt those nonlinear interaction phenomena.

Flexural-Torsional Post Critical Behavior of a Cantilever Beam Dynamically Excited: Theoretical Model and Experimental Tests

The 3D dynamics of a cantilever beam undergoing large displacements under a sinusoidally varying, concentrated, vertical force at its free end are analyzed in this paper. The Partial Differential Equations (PDEs) of the motion are obtained by using the Principle of Virtual Power. Then a reduced 4 degrees-of-freedom model is obtained using, in a Galerkin approximation, four eigenfunctions of the linearized model. The obtained four Ordinary Differential Equations (ODEs) of the motion are expanded by means of a 3rd order Multiple Time Scales perturbation technique to obtain Amplitude and Phase Modulation Equations (APMEs). The role of the inertial-elastic nonlinear terms, responsible for the coupling of the mass matrix, and of the viscous-elastic nonlinear terms, both usually neglected in the literature, is discussed. A path following procedure applied to the APMEs is used to describe the global dynamical behavior in the plane of the excitation control parameters. The results obtained using the 4 d.o.f. analytical model are compared with those of an experimental aluminium model of the cantilever. The regions of instability of the 1-modal planar solution, in which the nonlinear modal coupling excites out of plane and/or torsional components, are studied.Copyright

EXPERIMENTAL DYNAMICS OF A HANGING CABLE CARRYING TWO CONCENTRATED MASSES

The dynamics of a massless hanging cable with two heavy masses is examined experimentally. The experiment is also a model for a three-dimensional four-bar linkage or two coupled spherical pendulums. The system is subjected to a periodic in-phase and out-of-phase vertical motion of the hanging points. The regions of instability of the planar motion in which the system exhibits prechaotic and chaotic behavior are described by means of the Fourier transform, probability density function, and autocorrelation function. Charts of behavior in the frequency-amplitude excitation parameter plane show regions of periodic, quasiperiodic, and chaotic motions. In the chaotic regime a reconstruction from a scalar time series of the global properties of the attractor is done by means of the delay map technique. An analytical model is developed to compare analytical and numerical results. The structure of the experimental global attractor suggests a Shil’nikov model for the transition to chaotic behavior.