Papers by Leonid Manevitch
Volume 1: 23rd Biennial Conference on Mechanical Vibration and Noise, Parts A and B, 2011
We propose a simple analytical description of beating with complete energy transfer in terms of p... more We propose a simple analytical description of beating with complete energy transfer in terms of periodic non-smooth functions characterizing amplitude ratio of weakly coupled nonlinear oscillators and their phase shift. Such a description is similar to that of a vibro-impact process. We consider subsequently two weakly coupled con- servative, damped and forced nonlinear oscillators. It is shown that two dynamical
CISM International Centre for Mechanical Sciences, 2010
We present an adequate analytical approach to the description of nonlinear vibration with strong ... more We present an adequate analytical approach to the description of nonlinear vibration with strong energy exchange between weakly coupled oscillators and oscillatory chains. The fundamental notion of the limiting phase trajectory (LPT) corresponding to complete energy exchange is introduced. In certain sense this is an alternative to the nonlinear normal mode (NNM) characterized by complete energy conservation. Well-known approximations based on NNMs turn out to be valid for the case of weak energy exchange, and the proposed approach can be used for the description of nonlinear processes with strong energy exchange between weakly coupled oscillators or oscillatory chains. Such a description is formally similar to that of a vibro-impact process and can be considered as starting approximation when dealing with other processes with intensive energy transfer. At first we propose a simple analytical description of vibrations of nonlinear oscillators. We show that two dynamical transitions occur in the system. First of them corresponds to the bifurcation of anti-phase vibrations of oscillators. And the second one is caused by coincidence of LPT with separatrix dividing two stable stationary states and leads to qualitative change in both phase and temporal behavior of the LPT (in particular, temporal dependence of the amplitude becomes resembling that for vibro-impact vibrations). Next problem under consideration relates to intensive intermodal exchange in the periodic nonlinear systems with finite (n>2) number of degrees of freedom. We consider two limiting cases. If the number of particles is not large enough, the energy exchange between nonlinear normal modes in two-dimensional integral manifolds is considered. When the number of the particles increases the energy exchange between neighbor integral manifolds becomes important that leads to formation of the localized excitations resembling the breathers in the one-dimensional continuum media.
Foundations of Engineering Mechanics, 2002
Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2003
ABSTRACT We study the dynamics of a semi-infinite linear chain of particles that is weakly couple... more ABSTRACT We study the dynamics of a semi-infinite linear chain of particles that is weakly coupled to a strongly nonlinear oscillator at its free end. We analyze families of localized standing waves situated inside the lower or upper attenuation zones of the linear chain, corresponding to energy predominantly confined in the nonlinear oscillator. These families of standing waves are generated due to resonant interactions between the chain and the nonlinear attachment. A scenario for the realization of energy pumping phenomena in the system under consideration is discussed, and is confirmed by direct numerical simulations of the chain-attachment dynamic interaction.
Journal of Vibration and Acoustics, 2004
We examine passive energy pumping in a system of damped coupled oscillators. This is a one-way, p... more We examine passive energy pumping in a system of damped coupled oscillators. This is a one-way, passive and irreversible energy flow from a linear main system to a nonlinear attachment that acts, in essence, as a nonlinear energy sink (NES). Energy pumping is caused by 1:1 resonance captures on resonant manifolds of the damped systems. We show that the NES is capable of absorbing significant portions of the energies generated by transient, broadband external excitations. By performing a series of numerical simulations we confirm that the energy dependence of the nonlinear normal modes (NNMs) of the underlying undamped, unforced system determines, in essence, the resonance capture and energy pumping dynamics in the corresponding damped system. We present numerical simulations of single-and multi-mode energy pumping, that involve isolated resonance captures or resonance capture cascades, respectively. In addition, we discuss methodologies for enhancing the nonlinear energy pumping phenomenon by properly selecting the system parameters. The described technique of passively localizing and locally eliminating externally induced energy provides a new paradigm for vibration and shock isolation of mechanical oscillators.
Foundations of Engineering Mechanics, 2010
ABSTRACT Discrete systems comprising an infinite number of particles are rather challenging for a... more ABSTRACT Discrete systems comprising an infinite number of particles are rather challenging for a formulation of tractable models. The simplifications related to a low number of the degrees of freedom or normal modes cannot normally be implemented directly. Transition to a continuum and to a description in terms of partial differential equations may be involved and normally requires a number of assumptions; each of them should be carefully checked. In particular, for different regimes of motion of the discrete system one can obtain rather different continuous approximations. On the other side, such systems are extremely important both in mechanics (for modeling arrays of structural elements etc.) and in physics. For the study of the relationship between macroscopic properties and the microscopic structure in classical physics, the models of this sort are the best possible.
Foundations of Engineering Mechanics, 2010
The necessary precondition for the formulation of tractable models of mechanics is an adequate un... more The necessary precondition for the formulation of tractable models of mechanics is an adequate understanding of the main notions and concepts involved in the theoretical considerations. The lack of such understanding jeopardizes efficient scientific development. This is clearly seen in the historical context – the way to formation of mechanics was rather tortuous.
Foundations of Engineering Mechanics, 2010
ABSTRACT An asymptotic approach is rather efficient when dealing with the theory of oscillations,... more ABSTRACT An asymptotic approach is rather efficient when dealing with the theory of oscillations, since one often can figure out a number of relatively simple limit cases which can be efficiently treated and completely understood. Such an approach allows the deepest possible simplification but preserves the most significant features of dynamical behavior. At the same time, one can use the expansions by small parameters characterizing the deviation of the system from the tractable limit case.
Dynamics of Vibro-Impact Systems, 1999
Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2005
ABSTRACT We present analytical and numerical study of short wavelength breathers in the system of... more ABSTRACT We present analytical and numerical study of short wavelength breathers in the system of asymmetric nonlinear oscillators coupled by stretched weightless beam. The study is focused on modulations of the nonlinear normal mode with shortest wavelength. A small parameter is introduced as ratio of distance between the particles in the chain and characteristic wavelength of modulation. It is shown that in main asymptotic approach the problem can be reduced to Nonlinear Schrödinger Equation (NSE). Breathers (envelope solitons) are localized oscillating solutions of this equation. We reveal the stability and high mobility of such solutions in their numerical study based on integrating of exact discrete equations of motion. We study breathers, localized on chain ends, and radiation of breathers from chain ends, induced by variable external force.
Volume 1: 20th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2005
ABSTRACT We describe nonlinear dynamics of essentially non-homogeneous system of coupled oscillat... more ABSTRACT We describe nonlinear dynamics of essentially non-homogeneous system of coupled oscillators using the technique of multi-scale expansions. The resonant mechanism of passive energy transfer from a massive oscillator into a light auxiliary one provides essential reduction of energy of the massive oscillator. Proposed active control algorithm increases noticeably the efficiency of energy transfer in comparison with passive control. The important advantage of this algorithm is that it provides a small consumption of energy. It is expedient to use our algorithm in the certain limits of parameters determined in the paper. This algorithm provides a necessary level of reduction of oscillator amplitude in a wide range of external perturbations.
Volume 1: 23rd Biennial Conference on Mechanical Vibration and Noise, Parts A and B, 2011
ABSTRACT We introduce a novel type of the nonlinear energy sink (NES) designed as an eccentric ma... more ABSTRACT We introduce a novel type of the nonlinear energy sink (NES) designed as an eccentric mass rotating within a horizontal plane. The gravity is not a factor here, therefore such a rotator has no eigenfrequency and can inertially couple and resonate with any mode of the primary system. The dynamics of the system consisting of a primary linear oscillator and the eccentric rotator is rich beyond expectations and features multiple resonances and chaotic modes. A numerical study shows that the system, when subject to high impulsive loads, inevitably enters a 1:1 resonance that enables highly efficient targeted energy transfer from the primary mass to the NES. The results of an experimental investigation are in good agreement with the analytical and numerical estimates.
Springer Series in Synergetics, 1998
International Journal of Non-Linear Mechanics, 2011
We present a novel type of an energy trap providing targeted energy transfer in a system of weakl... more We present a novel type of an energy trap providing targeted energy transfer in a system of weakly coupled pendulums. Our approach is based on the analogy we revealed between the behavior of two weakly coupled classical parametric pendulums (in linear approximation) and the nonadiabatic Landau-Zener tunneling in a two-state quantum system. This analogy leads us to the prediction of
Solid Mechanics and its Applications, 2005
ABSTRACT
Foundations of Engineering Mechanics, 2002
Solid Mechanics and its Applications, 2005
We analyze energy pumping phenomena in a linear periodic substructure, weakly coupled to an essen... more We analyze energy pumping phenomena in a linear periodic substructure, weakly coupled to an essentially nonlinear attachment. By energy pumping we denote passive, one-way, irreversible transfer of energy from the linear substructure to the nonlinear attachment. As a specific application the dynamics of a system of linear coupled oscillators with a nonlinear end attachment is examined. Both theoretical and experimental results are discussed, including resonance capture cascades, i.e., a series of energy pumping phenomena occurring at different frequencies, with sudden lower frequency transitions between sequential events. The observed multifrequency energy pumping cascades are particularly interesting from a practical point of view, since they indicate that nonlinear attachments can be designed to resonate and extract energy from an a priori specified set of modes of a linear structure, in compatibility with the design objectives.
Volume 5: 19th Biennial Conference on Mechanical Vibration and Noise, Parts A, B, and C, 2003
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Papers by Leonid Manevitch