Angelo Di Egidio

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.

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.

Comparison between the seismic response of 2D and 3D models of rigid blocks

A three-dimensional rigid body on the shape of a parallelepiped is modelled in order to rock on a side or a vertex of the base, in order to evaluate the seismic response of rigid blocks lying on a horizontal support. The center of mass of the body is considered as eccentric with respect to its geometric center. As seismic input, three Italian recorded accelerograms, with different spectral content, are used. The study is mainly conducted to highlight the differences between the seismic response of 2D and 3D models of rigid blocks, with the aim to understand if, in some cases, the use of the 3D model of rigid block is required to obtain safer results. In fact, the outcomes show that in some ranges of the geometrical and mechanical parameters that characterize the excitation and the body, a two-dimensional model, which is not able to consider the 3D rocking on a vertex, can provide unsafe results. In particular, it is found that the overturning process of the three-dimensional block can occur under excitations which are lower than those which overturn a corresponding two-dimensional block.

Seismic Protection of Monolithic Objects of Art Using a Constrained Oscillating Base

The model of rigid block is well known in literature. In the past, several papers analyzed the behaviour of rigid blocks under different kind of excitations because many monolithic objects of art, such as statues, obelisks and fountains, subject to earthquake excitation, can be modelled as rigid blocks. In [Shenton & Jones, 1991] a general bi-dimensional formulation of the rigid block has been obtained and rocking and slide-rock approximated conditions have been written. More recently this model has been used to describe the behaviour of monolithic bodies subject to base excitations as a one-sine pulse excitation in [Zhang & Makris, 2001; Makris & Black, 2004, Kounadis 2010] and earthquake excitation in [Agbabian et al., 1988; Pompei et al, 1998; Taniguchi, 2002]. Almost all the papers on rigid blocks subject to base excitation focus their attention on symmetric rigid bodies. Only a few papers concern non-symmetric rigid bodies that, usually, represent objects of art better than symmetric rigid blocks. In [Boroscheck & Romo, 2004] the influence of the eccentricity of the centre of mass on the motion of the system has been studied. In [Purvance, 2005; Purvance et al., 2008] an analytical and experimental estimation of overturning events under seismic excitations has been carried out, both for symmetric and non-symmetric rigid bodies. In particular, in [Zhang & Makris, 2001] for a one-sine pulse excitation and in [Purvance, 2005; Purvance et al., 2008] for seismic excitation, the existence of survival regions that lie above the PGA (Peak Ground Acceleration) associated with the first overturning occurrence have been shown. In recent years, methods to reduce the effects of seismic excitation on art objects have been studied in some papers. In [Fujita et al, 2008 ] a critical excitation problem for a rigid block subjected to horizontal and vertical simultaneous base inputs is considered. In [Vestroni & Di Cinto, 2000] a base isolation system has been used to protect statues from seismic effects. The work of art has been modeled through an equivalent elastic beam. In [Calio & Marletta, 2003] the same problem has been analyzed, but the art object has been modeled as a symmetric rigid block simply supported on an oscillating base connected to the ground by a visco-elastic device. The sliding of the body is prevented by special seismic restraints. These analyses have shown the effectiveness of the isolation system and the role of many parameters. To make things more realistic, in [Contento & Di Egidio, 2009], the model presented in [Calio & Marletta, 2003] has been enriched considering also the eccentricity of the centre of mass of the rigid body and the presence of security stops, able to prevent the breaking of the isolation device by limiting

Linear static behavior of curved beams coupled with strings representing also fiber-reinforced masonry arches

A linear elastic analytical model of curved beam with constant curvature coupled with curved strings has been developed. To model the connections between the strings and the beam, a continuous distribution of tangential and orthogonal linear elastic springs have been introduced. To evaluate the behavior of the system, a numerical integration of the field equations has been performed. A parametric analysis, capable of pointing out the influence on the behavior of the system of both the stiffness of the tangential connection between strings and beam and of the geometrical characteristic of the beam, has been performed. The results of the proposed model have been compared with those obtained from a finite element model. They show that the strings, in some cases, do not work as expected.

On the use of reinforcing layers to improve the static behaviour of arches or shells with single curvature

A linear elastic model constituted by two interacting concentric arches is developed. The sensitivity of the behaviour of the system to its mechanical and geometrical parameters is studied to individuate the fundamental ones. The analysis is extended to the nonlinear field by means of computational finite element models. Another objective of the research is to specifically analyse the interaction between the two arches when there is a large difference in stiffness. A practical case is represented by the retrofitting of a masonry vault with a single curvature (barrel vaults) with a reinforcing concrete layer that, in several cases, resulted in the detachment of some masonry blocks. Finally, on the basis of the results of the nonlinear analyses, which qualitatively confirmed the results of the linear ones, a slightly innovative approach to the use of the reinforcing layers is proposed.

Seismic Behaviour of Monolithic Objects: A 3D Approach

Despite the many progresses done in the modelling of rigid blocks, the grounding work for most of the research in this field remains [1], where a 2D model of the rigid block is obtained and the rocking and slide-rocking approximated conditions are written. Following papers on the dynamics of rigid bodies can be divided in two main groups, according to the kind of excitation used: earthquake excitation or sine-type pulse excitation (mainly one-sine). To the first group belong [2-5], in the second one, [6-10] can be found. In time, models of rigid blocks, very useful in many research fields, have been increased in complexity. Recently, for instance, sliding phenomena and the eccentricity of the center of mass have been considered (see [3, 11]). Some papers have been focused to specific problems, for example in [12] the behavior of two stacked rigid blocks has been considered, whereas in [13, 14] the attention has been pointed to blocks on flexible foundations. The dynamics and control of 2D blocks have also been analyzed in the framework of the bifurcation theory in [15, 16, 17].