Dynamic instabilities in coupled oscillators induced by geometrically nonlinear damping (original) (raw)
Related papers
Archive of Applied Mechanics, 2014
We study the 1:3 resonant dynamics of a two degree-of-freedom (DOF) dissipative forced strongly nonlinear system by first examining the periodic steady-state solutions of the underlying Hamiltonian system and then the forced and damped configuration. Specifically, we analyze the steady periodic responses of the two DOF system consisting of a grounded strongly nonlinear oscillator with harmonic excitation coupled to a light linear attachment under condition of 1:3 resonance. This system is particularly interesting since it possesses two basic linearized eigenfrequencies in the ratio 3:1, which, under condition of resonance, causes the localization of the fundamental and third-harmonic components of the responses of the grounded nonlinear oscillator and the light linear attachment, respectively. We examine in detail the topological structure of the periodic responses in the frequency-energy domain by computing forced frequency-energy plots (FEPs) in order to deduce the effects of the 1:3 resonance. We perform complexification/averaging analysis and develop analytical approximations for strongly nonlinear steady-state responses, which agree well with direct numerical simulations. In addition, we investigate the effect of the forcing on the 1:3 resonance phenomena and conclude our study with the stability analysis of the steady-state solutions around 1:3 internal resonance, and a discussion of the practical applications of our findings in the area of nonlinear targeted energy transfer.
Physica D: Nonlinear Phenomena, 2012
We study a peculiar damped nonlinear transition of a system of two coupled oscillators into a state of sustained nonlinear resonance scattering. This system consists of a grounded, weakly damped linear oscillator attached to a light, weakly damped oscillator with essential (nonlinearizable) stiffness nonlinearity of the third degree, and linear or nonlinear damping. We find that under specific forcing conditions the damped response of this system locks into a damped, non-resonant transition resembling continuous resonance scattering, whereby the transient damped dynamics closely follows an impulsive orbit manifold of the dynamics in the frequency-energy plane. This manifold is formed by a countable infinity of periodic orbits and an uncountable infinity of quasi-periodic orbits of the underlying Hamiltonian system, with each of these orbits representing the response of the Hamiltonian system being initially at rest and forced by an impulse applied to the linear oscillator. Hence, the damped transitions reported here appear to lock in sustained resonance scattering from a countable infinity of periodic orbits along the impulsive orbit manifold. Such transitions represent an anti-resonance state, where the dynamics is farthest away from resonance. We conjecture that such transitions are only made possible by the essential (nonlinearizable) stiffness nonlinearity of the nonlinear attachment and cannot be realized in linearizable nonlinear dynamics where resonance captures prevent sustained resonance scattering. Our findings are supported by numerical, analytical and experimental results.
Non-linear dynamics of a system of coupled oscillators with essential stiffness non-linearities
International Journal of Non-Linear Mechanics, 2004
We study the resonant dynamics of a two-degree-of-freedom system composed of a linear oscillator weakly coupled to a strongly non-linear one, with an essential (non-linearizable) cubic sti ness non-linearity. For the undamped system this leads to a series of internal resonances, depending on the level of (conserved) total energy of oscillation. We study in detail the 1:1 internal resonance, and show that the undamped system possesses stable and unstable synchronous periodic motions (non-linear normal modes-NNMs), as well as, asynchronous periodic motions (elliptic orbits-EOs). Furthermore, we show that when damping is introduced certain NNMs produce resonance capture phenomena, where a trajectory of the damped dynamics gets 'captured' in the neighborhood of a damped NNM before 'escaping' and becoming an oscillation with exponentially decaying amplitude. In turn, these resonance captures may lead to passive non-linear energy pumping phenomena from the linear to the non-linear oscillator. Thus, sustained resonance capture appears to provide a dynamical mechanism for passively transferring energy from one part of the system to another, in a one-way, irreversible fashion. Numerical integrations conÿrm the analytical predictions.
Journal of Vibration and Acoustics, 2013
This report describes the impulsive dynamics of a system of two coupled oscillators with essential (nonlinearizable) stiffness nonlinearity. The system considered consists of a grounded weakly damped linear oscillator coupled to a lightweight weakly damped oscillating attachment with essential cubic stiffness nonlinearity arising purely from geometry and kinematics. It has been found that under specific impulse excitations the transient damped dynamics of this system tracks a high-frequency impulsive orbit manifold (IOM) in the frequency-energy plane. The IOM extends over finite frequency and energy ranges, consisting of a countable infinity of periodic orbits and an uncountable infinity of quasi-periodic orbits of the underlying Hamiltonian system and being initially at rest and subjected to an impulsive force on the linear oscillator. The damped nonresonant dynamics tracking the IOM then resembles continuous resonance scattering; in effect, quickly transitioning between multiple r...
Chaos, Solitons & Fractals, 2005
Energy transfer in 2 DOF systems comprised of linear oscillator with unit mass and strongly nonlinear oscillator with multiple states of equilibrium and relatively small mass is investigated by analytic and numerical methods. We demonstrate that the process of energy transfer is governed by structure of damped nonlinear normal modes of the system. Various dynamical regimes dependent on the system parameters are revealed and conditions for efficient energy pumping to the strongly nonlinear attachment are formulated. Possibility of transient chaotic response is investigated.
Dynamics of Nonlinear Oscillators under Simultaneous Internal and External Resonances
1998
An analysis is presented for a class of two degree of freedom weakly nonlinear oscillators, with symmetric restoring force. Conditions of one-to-three internal resonance and subharmonic external resonance of the lower vibration mode are assumed to be satisfied simultaneously. As a consequence, the second vibration mode may also be under the action of external primary resonance. Initially, a set of slow-flow equations is derived, governing the amplitudes and phases of approximate long time response of these oscillators, by applying an asymptotic analytical method. Determination of several possible types of steady-state motions is then reduced to solution of sets of algebraic equations. For all these solution types, appropriate stability analysis is also performed. In the second part of the study, this analysis is applied to an example mechanical system. First, a systematic search is performed, revealing effects of system parameters on the existence and stability properties of periodic motions. Frequency-response diagrams are presented and attention is focused on understanding the evolution and interaction of the various solution branches as the external forcing and nonlinearity parameters are varied. Finally, numerical integration of the equations of motion demonstrates that the system exhibits quasiperiodic or chaotic response for some parameter combinations.
Dynamics of coupled linear and essentially nonlinear oscillators with substantially different masses
Journal of Sound and Vibration, 2005
The dynamics of a linear oscillator, coupled to an essentially nonlinear attachment of substantially lower mass, is investigated. The essential (nonlinearizable) nonlinearity of the attachment enables it to resonate with the oscillator, leading to energy pumping phenomena, e.g., passive, almost irreversible transfer of energy from the substructure to the attachment. Feasibility of this process for possible applications depends on relative mass of the attachment, the obvious goal being to minimize it while preserving the efficiency of the pumping. Two different models of the attachment coupled to the main single-degree-of-freedom body are proposed and analyzed both analytically and numerically. It is demonstrated that efficient energy pumping may be obtained for a rather small value of the attachment mass. Two mechanisms of energy pumping in the system under consideration are revealed. The first one is similar to previously studied resonance capture; a novel analytic framework allowing explicit account of the damping is proposed. The second mechanism is related to nonresonant excitation of high-frequency vibrations of the attachment. Both mechanisms are demonstrated numerically for a model consisting of a linear chain with a nonlinear attachment. r
Nonlinear Dynamics, 2007
We study the dynamics of a system of coupled linear oscillators with a multi-DOF end attachment with essential (nonlinearizable) stiffness nonlinearities. We show numerically that the multi-DOF attachment can passively absorb broadband energy from the linear system in a one-way, irreversible fashion, acting in essence as nonlinear energy sink (NES). Strong passive targeted energy transfer from the linear to the nonlinear subsystem is possible over wide frequency and energy ranges. In an effort to study the dynamics of the coupled system of oscillators, we study numerically and analytically the periodic orbits of the corresponding undamped and unforced hamiltonian system with asymptotics and reduction. We prove the existence of a family of countable infinity of periodic orbits that result from combined parametric and external resonance interactions of the masses of the NES. We numerically demonstrate that the topological structure of the periodic orbits in the frequency-energy plane of the hamiltonian system greatly influences the strength of targeted energy transfer in the damped system and, to a great extent, governs the overall transient damped dynamics. This work may be regarded as a contribution towards proving the efficacy the utilizing essentially nonlinear attachments as passive broadband boundary controllers.
Out-of-unison resonance in weakly nonlinear coupled oscillators
Proceedings. Mathematical, physical, and engineering sciences / the Royal Society, 2015
Resonance is an important phenomenon in vibrating systems and, in systems of nonlinear coupled oscillators, resonant interactions can occur between constituent parts of the system. In this paper, out-of-unison resonance is defined as a solution in which components of the response are 90° out-of-phase, in contrast to the in-unison responses that are normally considered. A well-known physical example of this is whirling, which can occur in a taut cable. Here, we use a normal form technique to obtain time-independent functions known as backbone curves. Considering a model of a cable, this approach is used to identify out-of-unison resonance and it is demonstrated that this corresponds to whirling. We then show how out-of-unison resonance can occur in other two degree-of-freedom nonlinear oscillators. Specifically, an in-line oscillator consisting of two masses connected by nonlinear springs-a type of system where out-of-unison resonance has not previously been identified-is shown to ha...