R. Alaggio

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.

Characterizing bifurcations and classes of motion in the transition to chaos through 3D-tori of a continuous experimental system in solid mechanics

Abstract The dynamics of a continuous experimental system exhibiting quasiperiodic transition to chaos has been investigated. An elastic cable/mass system hanging at in-phase vertically moving supports is considered, with vibrational parameters in the neighbourhood of external and internal resonance conditions adjusted to be able to introduce three-torus dynamics. Measurements of the nonregular response are made in frequency ranges including primary resonance of the first symmetric in-plane mode of the cable. Quantitative characterisation of regular and nonregular attractors and of the configuration variables involved in the motion is made by means of delay-embedding technique and proper orthogonal decomposition of spatio-temporal flow. Attention is devoted to analyse in-depth the scenario of transition to chaos exhibited by the model. Steady quasiperiodic motions on three-tori, partial and full phase-locking of the flow, and chaotic attractors are observed. In spite of the dynamics variedness, the systematic characterisation of the response allows us to set the bifurcation behaviour of the experimental system in the framework of theoretic/numeric transition to chaos through three-dimensional tori.

Spatio-temporal dimensionality in the overall complex dynamics of an experimental cable/mass system

An experimental model of an elastic cable/mass hanging at in-phase or out-of-phase vertically moving supports is considered. System parameters are adjusted to produce two different conditions of multiple internal resonance. Nonregular dynamics are analyzed in various frequency ranges including meaningful external resonance conditions. Attention is devoted to characterization of system dimensionality in terms of both time and spatial complexity. The aims of this paper are (i) to give a general overview of the richness and robustness of different (quasiperiodic and homoclinic) bifurcation scenarios to chaos in various regions of the control parameter space, (ii) to characterize steady nonregular response through delay-embedding technique for attractor reconstruction and proper orthogonal decomposition of spatio-temporal flow and (iii) to identify spatial configuration variables (experimental eigenfunctions) contributing mostly to nonregular dynamics, thus obtaining hints about possible reduced models for reproducing complex regimes. System dimensionality will be evaluated both by relating the dimension of attractors to the dimension of the linear phase space, and from the dominating proper orthogonal modes.

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

The Valle Castellana Twin-Arch Bridge: Dynamical Tests, Identification, Seismic Performances

The assessment of the structural conditions of existing structures and the evaluation of their seismic performances are usually conducted by means of numerical models. Differently than in the case of a new structure when the construction following the design phase should be done trying to reproduce, in the best way, the design assumptions, assessment analysis requires the determination of a numerical model able to accurately reproduce the actual behavior of the existing construction. This latter issue leads to an inverse problem, whose difficulty is always higher than the direct design problem. In this paper, model updating strategies based on dynamic data are applied to a class of six reinforced concrete twin arch bridges located in Provincia di Teramo, Italy, to assess their structural performance under seismic actions. The attention will be mainly focused on the Valle Castellana bridge. The case study has been selected because an apparently strange dynamic behavior emerged in a dynamic campaign carried out on 2002–2003 on this bridge. The peculiar behavior can be linked to a known phenomenon possibly affecting initially curved, symmetric structures, when the initial symmetry is lost for different possible reasons. In addition, on 2009, the Valle Castellana bridge suffered for a sudden settlement of one of the supports, and restoration works were needed to reconstruct the parts subject to failure. Based on the results of dynamic tests carried out on 2011, a calibrated finite element model was determined and used to assess the actual structural behavior of the bridge. A possible structural improvement of its seismic performance based on the validated numerical model is also discussed.

Modal Identification of Superconducting Magnetic Levitating Bogie

A novel superconducting magnetic levitation transportation systems has been proposed at University of L’Aquila, Italy. The bogie floats due to a passive, self-balancing interaction between high temperature superconducting skaters on board and permanent magnets on the track, in all phases of motion, zero speed included. A scaled superconducting skater has been statically tested measuring the repulsive-attractive magnetic forces varying, in a controlled way, the distance between the skater and the track. A non linear hysteretic characteristic curve has been identified averaging a set of suitable measures. In a first step, considering the thinness of the hysteretic cycles, the characteristic curve has been simplified in a non linearly elastic one. On the same time the equivalent tangent stiffness of such a curve has been identified, knowing the geometry and the mass characteristics of the bogie, by an experimental modal analysis conducted in operational conditions. A companion numerical model of the system has been introduced to forecast the working conditions with particular attention to dynamic behavior.

Modal and Structural Identification of a Skew, Cable Stayed, Arch Bridge

After the construction of a bridge, different problem could arise demanding for an evaluation of the structural performances. In case of a complex structure, the use of a numerical model without an experimental validation is strongly discouraged.

Nonlinear Forced Dynamics of Planar Arches

The nonlinear dynamics of planar elastic arches, under resonant vertical harmonic tip force and different sag-to-span ratios, are considered. An analytical model based on the polar continuum, curved rod theory, is formulated. Different values of initial curvature are considered, ranging from non-shallow to shallow conditions. The nonlinear change in curvature is expressed in terms of displacement components. The hypothesis of vanishing axial strain is assumed when dealing with non-shallow cases, while the Mettler theory, based on a constant strain, is usually used for shallow arches. The PDEs of the motion obtained through the extended Hamilton principle, are projected on the reduced basis constituted by the two first linear modes or analogous meaningful functions. Regions of instability of the one-mode solution are numerically detected, and coupled regular and non-regular motions are described using standard complexity indicators. Experimental tests are realized on two companion laboratory steel prototypes of arches, in order to compare and validate the results of the prevision of the analytical model. The behavior charts of the analytical and experimental problems and the nonlinear frequency-response curves show good agreement in a wide range of amplitude and frequency of the external excitation. The 2D model recently proposed by the authors seems to be the only adequate when dealing with real arches.

Forced 3D nonlinear dynamics of a hanging cable under multiple resonance conditions

The forced, nonlinear, 3D dynamics of an elastic cable is analyzed by means of a reduced 4 d.o.f. model, already obtained several years ago by some of the Authors of this paper. The system is analyzed in the case of multiple internal resonance conditions and a 1:1 external primary resonance condition. The reduced model, because of a strong intrinsic symmetry due to the fact that anti-symmetric in-plane and out-of-plane modes have the same natural frequency (Irvine’s theory), is in principle not able to catch some interesting classes of motion, such as ballooning, which on the other hand have been observed in experimental tests. In the present paper, an imperfection between the equations ruling the in-plane and out-of-plane components is introduced through an internal detuning, which simulates the slight difference between the frequencies of the two involved modes, which is plausible as a consequence of the initial curvature of the cable as well as obtainable through more refined analytical models. A discussion on the similarity and differences with the solutions previously obtained is presented. Regions of non-regular response in the excitation control parameter plane are located and ballooning trajectories are analyzed.

Assessment of the Structural Conditions of the San Clemente a Vomano Abbey

The simultaneous use of a Finite Element (FE) accurate modeling, dynamical tests, model updating and nonlinear analysis are used to describe the integrated approach used by the authors to assess the structural conditions and the seismic vulnerability of an historical masonry structure: the Abbey Church of San Clemente al Vomano, situated in the Notaresco territory (TE, Italy) commissioned by Ermengarda, daughter of the Emperor Ludovico II, and built at the end of IX century together with a monastery to host a monastic community. Dynamical tests “in operational conditions” and modal identification have been used to perform the FE model validation. Both a simple and direct method as the kinematic analysis applied on meaningful sub‐structures and a nonlinear 3D dynamic analysis conducted by using the FE model have been used to forecast the seismic performance of the Church.