Achille Paolone

Multiple Timescales Analysis for 1:2 and 1:3 Resonant Hopf Bifurcations

The postcritical behavior of a generaln-dimensional system around a resonant double Hopf bifurcation isanalyzed. Both cases in which the critical eigenvalues are in ratios of1:2 and 1:3 are investigated. The Multiple Scale Method is employedto derive the bifurcation equations systematically in terms of thederivatives of the original vector field evaluated at the criticalstate. Expansions of the n-dimensional vector of state variables andof a three-dimensional vector of control parameters are performed interms of a unique perturbation parameter ε, of the order ofthe amplitude of motion. However, while resonant terms only appear atthe ε3-order in the 1:3 case, they already arise at theε2-order in the 1:2 case. Thus, by truncating theanalysis at the ε3-order in both cases, first orsecond-order bifurcation equations are respectively drawn, the latterrequiring resort to the reconstitution principle. A two-degrees-of-freedom system undergoing resonant double Hopf bifurcations isstudied. The complete postcritical scenario is analyzed in terms of thethree control parameters and the asymptotic results are compared withexact numerical integrations for both resonances. Branches of periodicas well as periodically modulated solutions are found and theirstability analyzed.

Perturbation Methods for Bifurcation Analysis from Multiple Nonresonant Complex Eigenvalues

It is shown that the logical bases of the static perturbation method, which is currently used in static bifurcation analysis, can also be applied to dynamic bifurcations. A two-time version of the Lindstedt–Poincaré Method and the Multiple Scale Method are employed to analyze a bifurcation problem of codimension two. It is found that the Multiple Scale Method furnishes, in a straightforward way, amplitude modulation equations equal to normal form equations available in literature. With a remarkable computational improvement, the description of the central manifold is avoided. The Lindstedt–Poincaré Method can also be employed if only steady-state solutions have to be determined. An application is illustrated for a mechanical system subjected to aerodynamic excitation.

On the Reconstitution Problem in the Multiple Time-Scale Method

Higher-order multiple-scale methods for general multiparameter mechanical systems are studied. The role played by the control and imperfection parameters in deriving the perturbative equations is highlighted. The definition of the codimension of the problem, borrowed from the bifurcation theory, is extended to general systems, excited either externally or parametrically. The concept of a reduced dynamical system is then invoked. Different approaches followed in the literature to deal with reconstituted amplitude equations are discussed, both in the search for steady-state solutions and in the analysis of stability. Four classes of methods are considered, based on the consistency or inconsistency of the approach, and on the completeness or incompleteness of the terms retained in the analysis. The four methods are critically compared and general conclusions drawn. Finally, three examples are illustrated to corroborate the findings and to show the quantitative differences between the various approaches.

Elastodynamics of Nonshallow Suspended Cables: Linear Modal Properties

A mechanical model describing finite motions of nonshallow cables around the initial catenary configurations is proposed. An exact kinematic formulation accounting for finite displacements is adopted, whereas the material is assumed to be linearly elastic. The nondimensional mechanical parameters governing the motions of nonshallow cables are obtained via a suitable nondimensionalization, and the regions of their physically plausible values are portrayed. The spectral properties of linear unforced undamped vibrations around the initial static configurations are investigated via a Galerkin-Ritz discretization. A classification of the modes is obtained on the basis of their associated energy content, leading to geometric modes, elastostatic modes (with prevalent transverse motions and appreciable stretching), and elastodynamic modes (with prevalent longitudinal motion). Moreover, an extension of Irvine s model to moderately nonshallow cables is proposed to determine the frequencies and mode shapes in closed form.

On the Proper Form of the Amplitude Modulation Equations for Resonant Systems

The complex amplitude modulation equations of a discrete dynamicalsystem are derived under general conditions of simultaneous internal andexternal resonances. Alternative forms of the real amplitude and phaseequations are critically discussed. First, the most popular polar formis considered. Its properties, known in literature for a multitude ofspecific problems, are here proven for the general case. Moreover, thedrawbacks encountered in the stability analysis of incomplete motions(i.e. motions containing some zero amplitudes) are discussed as aconsequence of the fact the equations are not in standard normal form.Second, a so-called Cartesian rotating form is introduced, which makesit possible to evaluate periodic solutions and analyze their stability,even if they are incomplete. Although the rotating form calls for theenlargement of the space and is not amenable to analysis of transientmotions, it systematically justifies the change of variables sometimesused in literature to avoid the problems of the polar form. Third, amixed polar-Cartesian form is presented. Starting from the hypothesisthat there exists a suitable number of amplitudes which do not vanish inany motion, it is proved that the mixed form leads to standard formequations with the same dimension as the polar form. However, if suchprincipal amplitudes do not exist, more than one standard form equationshould be sought. Finally, some illustrative examples of the theory arepresented.

Multiple-Timescale Analysis for Bifurcation from a Multiple-Zero Eigenvalue

Multiple-zero bifurcation of a general multiparameter dynamic system is analyzed using the multiple-scale method and exploiting the close similarities with eigensolution analysis for defective systems. Because of the coalescence of the eigenvalues, the Jacobian matrix at the bifurcation is nilpotent. This entails using timescales withfractionalpowersoftheperturbationparameter.Thereconstitution methodisemployedto obtainanordinary differential equation of order equal to the algebraic multiplicity of the zero eigenvalue, in the unique unknown amplitude. When the algorithm is applied to a double-zero eigenvalue, Bogdanova ‐Arnold’ s normal form for the bifurcation equation is recovered. A detailed step-by-step algorithm is described for a general system to obtain the numerical coefe cients of the relevant bifurcation equation. The mechanical behavior of a nonconservative two-degree-of-freedom system is studied as an example.

Sensitivities and Linear Stability Analysis Around a Double-Zero Eigenvalue

A general, multiparameter system admitting a double-zero eigenvalue at a critical equilibrium point is considered. A sensitivity analysis of the critical eigenvalues is performed to explore the neighborhood of the critical point in the parameter space. Because the coalescence of the eigenvalues implies that the Jacobian matrix is defective (or nilpotent ), well-suited techniques of perturbation analysis must be employed to evaluate the eigenvalues and the eigenvector sensitivities. Different asymptotic methods are used, based on perturbations both of the eigenvalue problem and the characteristic equation. The analysis reveals the existence of a generic (nonsingular ) case and of a nongeneric (singular) case. However, even in the generic case, a codimension-1 subspace exists in the parameter space on which a singularity occurs. By the use of the relevant asymptotic expansions, linear stability diagrams arebuiltup, and different bifurcation mechanisms (divergence‐Hopf, doubledivergence, doubledivergence ‐Hopf, degenerateHopf )arehighlighted. The problem of e nding a uniqueexpression uniformly valid in the wholespaceis then addressed. It isfound that a second-degreealgebraic equation governsthebehaviorofthecriticaleigenvalues. It also permits clarie cation of the geometrical meaning of the unfolding parameters, which has been discussed in literature for the Takens ‐Bogdanova bifurcation. Finally, a mechanical system loaded by nonconservative forces and exhibiting a double-zero bifurcation is studied as an example.

Fano effect in the a-b plane of Nd{sub 1.96}Ce{sub 0.04}CuO{sub 4+y}: Evidence of phonon interaction with a polaronic background

Reflectance measurements in different Nd 1.96Ce0.04CuO41y samples with properly selected carrier concentrations provide firm evidence of Fano antiresonances for the four transverse optical Eu phonons in the a-b plane. An analysis of the Fano line shapes and of the dependence on temperature of the renormalized phonon frequencies allows us to determine the origin of the electronic continuum interacting with the phonons. Unlike in doped semiconductors, where the continuum is the free-carrier absorption band, here the continuum is provided by a polaron band at ;0.1 eV, present in most parent compounds of high-Tc superconductors. In the most doped sample, the polaron band softens as T decreases, thus indicating a delocalization of the polaronic carriers at low temperature. @S0163-1829~98!08501-4# In 1961, Fano quantitatively accounted for the so-called Fano antiresonance, i.e., the asymmetric line shape of the 2s2p 1 P resonance of He observed in electron inelastic scattering experiments. 1 In that work, Fano revised an earlier qualitative interpretation 2 of the spectra of He and other rare gases. Fano pointed out that such effects are expected whenever a set of discrete states is mixed with a continuous spectrum. Thereafter, ‘‘Fano profiles’’ have been observed in a number of spectra, including those where phonon discrete states interact with a continuum background due to free carriers. In high-critical-temperature superconductors ~HCTS’s !, phonon Fano profiles have been detected with the electric field of the radiation polarized along the c axis, either in 3‐6 YBa2Cu3O72d or in 7 Bi2Sr2CaCu2O8. Moreover, in the 400‐ 600 cm 21 region of the reflectivity spectra of Pb 2 Sr 2 LCu 3 O 8 ~L5Y , Dy, Eu, Nd, and Pr!, phonon line shapes have been reported to become asymmetric for increasing doping. 8 These results have been explained in terms of an interaction of Raman-active modes along the c axis, made infrared active by some symmetry-breaking potential, and an electronic continuum which develops with doping in the midinfrared. As far as we know, no evidence of Fano line shapes has been reported for the infrared-active phonons of the a-b plane. Here, indeed, the optical phonons are shielded by the carriers which form below Tc a fluid of superconducting pairs. In order to look for Fano resonances in the a-b plane of a cuprate, we have selected a strongly doped, nonmetallic system, where the existence of a considerable electron-phonon coupling is ensured by previous observations of polaronic effects in the a-b plane reflectivity. It is the case of Nd22xCexCuO42y ~NCCO!, where an electronic continuum detected at ;1000 cm 21 has been explained in terms of a polaron band d, whose strength increases with either x see ~Ref. 9! or y ~see Ref. 10!. The polaronic origin of the d band has been firmly established in Nd 2CuO42y ~NCO!. 10 Here it exhibits a fine structure, well explained in terms of an electron-phonon interaction. This structure has been shown to be related to some infrared-active vibrations ~IRAV’s !, observed in the far-infrared spectra of samples where carriers are either photoinduced or chemically induced. 11‐15 These IRAV’s add to the extended phonons of the undisturbed stoichiometric crystal, and appear in the far infrared as new peaks or sidebands of the phonons. 16,17 An infrared band at ;1000 cm 21 similar to that found in e-doped NCO has been observed also in h-doped, oxygen-enriched La2CuO41y ~LCO!, where both its line shape and temperature dependence are consistent with the absorption from a polaronic impurity state. 18 Such infrared evidence is confirmed by a broad spectrum of experimental techniques, ranging from extended x-ray-absorption fine structure, 19 to electron diffraction, 20 and to neutron scattering 21 and points toward the existence of polaron superstructures in superconducting cuprates as well as in nonsuperconducting perovskites. In the following it will be shown that asymmetric line shapes are found in Nd1.96Ce0.04CuO42y for the four optically active Eu phonons in the a-b plane. These asymmetric line shapes will be explained in detail in terms of a Fano interaction between the phonons and an electronic background. Contrarily to what is normally found in doped semiconductors, the electronic continuum is not given by the weak Drude term eventually present in these semiconducting compounds, but by the polaron d band peaked at

Damage evolution and debonding in hybrid laminates with a cohesive interfacial law

The hybridisation of fibres reinforced laminates, i.e., the combined use of two or more families of fibres, is an effective technique to achieve a pseudo-ductile response and overcome the inherent brittleness which limits the wider use of composite materials. In this paper, a one-dimensional analytical model for unidirectional hybrid laminates is derived. The model considers two elastic–brittle layers bonded together by a cohesive elasto–plastic–brittle interface. This formulation is applied to the study of the debonding and fracture of laminates under uniaxial loading and the results compared to experiments available from the open literature. This study shows that the proposed model provides a close fit to the experimental data and it is able to match accurately the crack patterns seen in the experiments. The model predicts four different failure mechanisms and is able to discriminate among them according to the geometrical and mechanical properties of the layers.

Anelastic spectroscopy study of the spin-glass and cluster spin-glass phases of La 2 − x Sr x CuO 4 ( 0.015 x 0.03 )

The anelastic spectra of