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In the past ** A.F. Vakakis** has collaborated on articles with

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AbstractForced localization in a periodic system consisting of an infinite number of coupled non-linear oscillators is examined. A “continuum approximation” is used to reduce the infinite set of ordinary differential equations of motion to a single approximate, non-linear partial differential equation. The structure of the propagation and attenuation zones of the linearized system is found to affect the non-linear localization. Harmonic excitations with general spatial distributions are considered and the localized responses of the chain are studied using exact and asymptotic techniques. Only certain classes of forcing distributions lead to spatial confinement of the forced responses, whereas other types of excitation give rise to spatially periodic or even chaotic harmonic motions of the chain. Systems with weak coupling between particles and/or strong non-linear effects have more profound localization characteristics. The theoretical predictions of the analysis are verified by direct numerical simulations of the equations of motion.

AbstractAn analytical procedure for the dynamic analysis of the unidirectional periodic isolator, consisting of n concentrated masses and n intermediate arbitrary blocks is developed. Complex polynomials depending on the four pole parameters of the mounts and on the frequency of excitation are introduced, to express analytical forms for the impedances and transmissibilities of the general system. By means of these polynomials, the frequency equation of the undamped isolator can be derived directly, for free or fixed boundary conditions. Application of the method was made with isolators consisting of masses and distributed elements of rubber with internal damping.

AbstractThe effect of a viscously damped dynamic absorber on the dynamic behaviour of a linear vibration system with many degrees of freedom is investigated. The dynamic absorber is connected to the roof of the primary system. In the sequence, optimal values for the parameters describing the behaviour of the dynamic absorber are determined, in order that the transmissibility of the composite system be minimized over the whole frequency range. As an application, respective types of anti-vibration mountings are proposed.

AbstractThe free oscillations of a strongly non-linear, discrete oscillator are examined by computing its “non-similar non-linear normal modes.” These are motions represented by curves in the configuration space of the system, and they are not encountered in classical, linear vibration theory or in existing non-linear perturbation techniques. The Mikhlin-Manevich asymptotic methodology is used for solving the singular functional equation describing the non-similar modes and approximate, analytical expressions are derived. For an oscillator with weak coupling stiffness and “mistuning,” both localized and non-localized modes are detected, occurring in small neighborhoods of “degenerate” and “global” similar modes of the “tuned” system. When strong coupling is considered, only non-localized modes were found to exist. An interesting result of this work is the detection of mode localization in the “tuned” periodic system, a result with no counterpart in existing theories on linear mode localization. As a check of the analytical results, numerical integrations of the equations of motion were carried out and the existence of the theoretically predicted non-similar modes was verified.

AbstractEquilibrium points, primary and secondary static bifurcation branches, and periodic orbits with their bifurcations of discrete systems under partial follower forces and no initial imperfections are examined. Equilibrium points are computed by solving sets of simultaneous, non-linear algebraic equations, whilst periodic orbits are determined numerically by solving 2- or 4-dimensional non-linear boundary value problems. A specific application is given with Ziegler's 2-DOF cantilever model. Numerous, complicated static bifurcation paths are computed as well as complicated series of periodic orbit bifurcations of relatively large periods. Numerical simulations indicate that chaotic-like transient motions of the system may appear when a forcing parameter increases above the divergence state. At these forcing parameter values, there co-exist numerous branches of bifurcating periodic orbits of the system; it is conjectured that sensitive dependence on initial conditions due to the large number of co-existing periodic orbits causes the chaotic-like transients observed in the numerical simulations.

AbstractNon-smooth time transformations are used to investigate strongly non-linear periodic free oscillations of a vibro-impact system with two degrees of freedom. Allowance for the boundary conditions at collision times enables the singularities induced by these transformations to be eliminated. The smoothed equations of motion turn out to be linear. Investigation of the periodic solutions reveals vibro-impact states with one- an two-sided collisions, including localized states (only one of the masses experiences collisions with stopping devices), and their bifurcation structure.

AbstractAlthough mechanical joints are integral parts of most practical structures, their modelling and their effects on structural dynamics are not yet fully understood. This represents a serious impediment to accurate modelling of the dynamics and to the development of reduced-order, finite element models capable of describing the effects of mechanical joints on the dynamics. In this work we provide an experimental study to quantify the non-linear effects of a typical shear lap joint on the dynamics of two structures: a beam with a bolted joint in its center; and a frame with a bolted joint in one of its members. Both structures are subjected to a variety of dynamical tests to determine the non-linear effects of the joints. The tests reveal several important influences on the effective stiffness and damping of the lap joints. The possibility of using Iwan models to represent the experimentally observed joint effects is discussed.

AbstractIn this work we show that it is possible to successfully apply the concept of nonlinear targeted energy transfer (TET) to seismic protection of structures; moreover, this passive strategy of seismic vibration control is found to be feasible and robust. We consider a three-story shear-frame structure, modeled as a three-degree-of-freedom system, subjected to four historic earthquakes as seismic excitation. Seismic mitigation is achieved by applying single or multiple nonlinear energy sinks (NESs) to the test structure. We study the performance and efficiency of the NESs through a set of evaluation criteria. First we consider a single vibro-impact NES (VI NES) applied to the top floor of the structure. In order to assess the robustness of the VI NES, the NES parameters are optimized for a specific seismic excitation (Kobe), and then tested against the three other earthquake records to demonstrate effectiveness of the NES for these cases as well. To further improve the effectiveness of the seismic mitigation, we then consider a system of two NESs—an NES with smooth nonlinearity at the top floor of the test structure and a VI NES at the bottom floor. We show that it is indeed possible to drastically reduce the structural seismic response (e.g., displacements, drifts, and accelerations) using this configuration.

AbstractThe free and forced motions of ordered and disordered layered systems are analyzed. The structure of propagation and attenuation zones (PZs and AZs) of the finite system depends on two non-dimensional parameters, v and τ. The parameter v is the ratio of the times of wave propagation at phase velocity through each layer, whereas parameter τ denotes the ratio of mechanical impedances of the materials forming the two layers. Systems with finite values of v and large or small values of τ are weakly coupled and possess narrow PZs and wide AZs. Depending on the value of v, a distinction is made between "degenerate" and "non-degenerate" PZs. The "degenerate" PZs are wider than "non-degenerate" ones. For small or large values of τ, the finite periodic system includes dense "clusters" of natural frequencies; when forced by a trapezoidal pulse, the maximum compressional force of the first arrival of the stress wave is localized close to the point of application of the excitation. For values of τ of order unity, this localization is eliminated. The effect on the free and forced response of disorder is then investigated. For a sufficiently strong disorder, a natural frequency shifts from a PZ to an AZ of the ordered system. This results in a spatial localization of the corresponding eigenfunction at the disorder. Finally, the transient responses of disordered and ordered systems are computed and compared.

AbstractWe study strongly nonlinear dynamical interactions between traveling waves propagating in a linear spring-mass chain with a strongly nonlinear, lightweight local attachment. We analyze the dynamics of this system by constructing a reduced model in the form of a strongly nonlinear integro-differential equation with inhomogeneous terms representing local and non-local interactions between the chain and the nonlinear attachment. Then we construct homoclinic and subharmonic Melnikov functions and prove the existence of chaotic motions and subharmonic periodic orbits in the combined chain-attachment system. In the limit of weak coupling between the particles of the chain we study the bifurcations that generate stable-unstable pairs of subharmonic motions. Generalizations of the methodology to a more general class of linear systems with local essentially nonlinear attachments are discussed. This work provides further evidence that the break of symmetry of an otherwise linear chain by a strongly nonlinear (even lightweight) attachment can give rise to complex (even chaotic) dynamics. The underlying dynamical mechanism of this complexity is nonlinear resonant energy transfer from the traveling waves to the nonlinear attachment. The presented results contribute towards the study of the dynamic and resonance interactions of waves propagating in extended media with strongly nonlinear local attachments.

AbstractThe concept of `non-linear normal mode' (NNM) is discussed. After providing some introductory definitions the applications of NNMs to vibration theory are considered. In particular, it is shown how this concept can be used to study forced resonances of non-linear systems and non-linear localisation of vibrational energy in symmetric systems. NNMs can provide a valuable tool for understanding certain essentially non-linear dynamic phenomena that have no counterparts in linear theory and that cannot be analysed by conventional linearised methods. Additional applications of NNMs to modal analysis, model reduction, vibration and shock isolation designs, and the theory of non-linear oscillators are also discussed.

AbstractThe Karhunen–Loeve (K–L) decomposition procedure is applied to a system of coupled cantilever beams with non-linear grounding stiffnesses and a system of non-linearly coupled rods. The former system possesses localized non-linear normal modes (NNMs) for certain values of the coupling parameters and has been studied in the literature using various asymptotic techniques. In this work, the K–L method is used to locate the regions of such localized motions. The method yields orthogonal modes that best approximate the spatial behaviour of the beams. In order to apply this method simultaneous time series of the displacements at several points of the system are required. These measurements are obtained by a direct numerical integration of the governing partial differential equations, using the assumed modes method. A two-point correlation matrix is constructed using the measured time-series data, and its eigenvectors represent the dominant K–L modes of the system; the corresponding eigenvalues give an estimate of the participations (energies) of these modes in the dynamics. These participations are used to estimate the dimensionality of the system and to identify regions of localized motion in the coupling parameter space. The same approach is applied to a system of non-linearly coupled rods. Through the comparison of system response reconstructions of the responses using a simple K–L mode and a number of physical modes, it is shown that the K–L modes can be used to create lower-order models that can accurately capture the dynamics of the original system.

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