Synthesis of optically active 6-substituted 2-(aminomethyl)chromans

Three novel, optically active, 6-substituted 2-(aminomethyl)chromans were synthesized from readily available chroman 2-carboxylic acid precursors. These chroman-containing primary amines are useful building blocks for the synthesis of chroman-derived pharmaceutical agents.     

Isolated resonances and nonlinear damping

We analyze isolated resonance curves (IRCs) in single-degree-of-freedom systems possessing nonlinear damping. Through the combination of singularity theory and the averaging method, the onset and merging of IRCs, which coincide to isola and simple bifurcation singularities, respectively, can be analytically predicted. Numerical simulations confirm the accuracy of the analytical developments. Another important finding of this paper is that we unveil a geometrical connection between the topology of the damping force and IRCs. Specifically, we demonstrate that extremas and zeros of the damping force correspond to the appearance and merging of IRCs. Considering a damping force possessing several minima and maxima confirms the general validity of the analytical result. It also evidences a very complex scenario for which different IRCs are created, co-exist and then merge together to form a super IRC which eventually merges with the main resonance peak.     

Nonlinear normal modes and band zones in granular chains with no pre-compression

We study standing waves (nonlinear normal modes—NNMs) and band zones in finite granular chains composed of spherical granular beads in Hertzian contact, with fixed boundary conditions. Although these are homogeneous dynamical systems in the notation of Rosenberg (Adv. Appl. Mech. 9:155–242, 1966), we show that the discontinuous nature of the dynamics leads to interesting effects such as separation between beads, NNMs that appear as traveling waves (these are characterized as pseudo-waves), and localization phenomena. In the limit of infinite extent, we study band zones, i.e., pass and stop bands in the frequency–energy plane of these dynamical systems, and classify the essentially nonlinear responses that occur in these bands. Moreover, we show how the topologies of these bands significantly affect the forced dynamics of these granular media subject to narrowband excitations. This work provides a classification of the coherent (regular) intrinsic dynamics of one-dimensional homogeneous granular chains with no pre-compression, and provides a rigorous theoretical foundation for further systematic study of the dynamics of granular systems, e.g., the effects of disorders or clearances, discrete breathers, nonlinear localized modes, and high-frequency scattering by local disorders. Moreover, it contributes toward the design of granular media as shock protectors, and in the passive mitigation of transmission of unwanted disturbances.     

Dynamic testing of nonlinear vibrating structures using nonlinear normal modes

Modal testing and analysis is well-established for linear systems. The objective of this paper is to progress toward a practical experimental modal analysis (EMA) methodology of nonlinear mechanical structures. In this context, nonlinear normal modes (NNMs) offer a solid theoretical and mathematical tool for interpreting a wide class of nonlinear dynamical phenomena, yet they have a clear and simple conceptual relation to the classical linear normal modes (LNMs). A nonlinear extension of force appropriation techniques is developed in this study in order to isolate one single NNM during the experiments. With the help of time-frequency analysis, the energy dependence of NNM modal curves and their frequencies of oscillation are then extracted from the time series. The proposed methodology is demonstrated using two numerical benchmarks, a two-degree-of-freedom system and a planar cantilever beam with a cubic spring at its free end.     

Energy Transfers in a System of Two Coupled Oscillators with Essential Nonlinearity: 1:1 Resonance Manifold and Transient Bridging Orbits

The purpose of this study is to highlight and explain the vigorous energy transfers that may take place in a linear oscillator weakly coupled to an essentially nonlinear attachment, termed a nonlinear energy sink. Although these energy exchanges are encountered during the transient dynamics of the damped system, it is shown that the dynamics can be interpreted mainly in terms of the periodic orbits of the underlying Hamiltonian system. To this end, a frequency-energy plot gathering the periodic orbits of the system is constructed which demonstrates that, thanks to a 1:1 resonance capture, energy can be irreversibly and almost completely transferred from the linear oscillator to the nonlinear attachment. Furthermore, it is observed that this nonlinear energy pumping is triggered by the excitation of transient bridging orbits compatible with the nonlinear attachment being initially at rest, a common feature in most practical applications. A parametric study of the energy exchanges is also performed to understand the influence of the parameters of the nonlinear energy sink. Finally, the results of experimental measurements supporting the theoretical developments are discussed. This study was carried out while the author was a postdoctoral fellow at the National Technical University of Athens and at the University of Illinois at Urbana-Champaign.     

Targeted energy transfers in vibro-impact oscillators for seismic mitigation

In the field of seismic protection of structures, it is crucial to be able to diminish ‘as much as possible’ and dissipate ‘as fast as possible’ the load induced by seismic (vibration-shock) energy imparted to a structure by an earthquake. In this context, the concept of passive nonlinear energy pumping appears to be natural for application to seismic mitigation. Hence, the overall problem discussed in this paper can be formulated as follows: Design a set of nonlinear energy sinks (NESs) that are locally attached to a main structure, with the purpose of passively absorbing a significant part of the applied seismic energy, locally confining it and then dissipating it in the smallest possible time. Alternatively, the overall goal will be to demonstrate that it is feasible to passively divert the applied seismic energy from the main structure (to be protected) to a set of preferential nonlinear substructures (the set of NESs), where this energy is locally dissipated at a time scale fast enough to be of practical use for seismic mitigation. It is the aim of this work to show that the concept of nonlinear energy pumping is feasible for seismic mitigation. We consider a two degree-of-freedom (DOF) primary linear system (the structure to be protected) and study seismic-induced vibration control through the use of Vibro-Impact NESs (VI NESs). Also, we account for the possibility of attaching to the primary structure additional alternative NES configurations possessing essential but smooth nonlinearities (e.g., with no discontinuities). We study the performance of the NESs through a set of evaluation criteria. The damped nonlinear transitions that occur during the operation of the VI NESs are then studied by superimposing wavelet spectra of the nonlinear responses to appropriately defined frequency – energy plots (FEPs) of branches of periodic orbits of underlying Conservative systems.     

Computing the endomorphism ring of an ordinary elliptic curve over a finite field

We present two algorithms to compute the endomorphism ring of an ordinary elliptic curve E defined over a finite field Fq. Under suitable heuristic assumptions, both have subexponential complexity. We bound the complexity of the first algorithm in terms of , while our bound for the second algorithm depends primarily on log|DE|, where DE is the discriminant of the order isomorphic to End(E). As a byproduct, our method yields a short certificate that may be used to verify that the endomorphism ring is as claimed.     

A modal separation measurement technique for broadband noise propagating inside circular ducts

A measurement technique which separates broadband noise propagating inside circular ducts into the acoustic duct modes is developed. The technique is also applicable to discrete frequency noise. The acoustic modes are produced by weighted combinations of the instantaneous outputs of microphones spaced around the duct circumference. The technique is compared with the cross spectral density approach presently available and found to have certain advantages, and disadvantages. Considerable simplification of both the new technique and the cross spectral density approach occurs when no correlation exists between different circumferential mode orders. The properties leading to uncorrelated modes and experimental tests which verify this condition are discussed. The modal measurement technique-is applied to the case of broadband noise generated by flow through a coaxial obstruction (nozzle or orifice) in a pipe. Different circumferential mode orders are shown to be uncorrelated for this type of noise source.     

On the functional form of a nonlinear vibration absorber

Due to the frequency-energy dependence of nonlinear oscillations, nonlinear dynamical absorbers present interesting properties for mitigating unwanted vibrations in mechanical systems. Unlike the tuned mass damper, the functional form of a nonlinear absorber is not known a priori and must be determined. This short note addresses this issue when a light-weight nonlinear absorber is attached to a nonlinear primary structure. Numerical simulations demonstrate that the determination of an adequate functional form may be directly linked to the frequency-energy dependence of the primary structure.     

An effective finite-element-based method for the computation of nonlinear normal modes of nonconservative systems

This paper addresses the numerical computation of nonlinear normal modes defined as two-dimensional invariant manifolds in phase space. A novel finite-element-based algorithm, combining the streamline upwind Petrov–Galerkin method with mesh moving and domain prediction–correction techniques, is proposed to solve the manifold-governing partial differential equations. It is first validated using conservative examples through the comparison with a reference solution given by numerical continuation. The algorithm is then demonstrated on nonconservative examples.