Qualitative analysis of a mechanical system of coupled nonlinear oscillators (original) (raw)

Dynamics of a pair of coupled nonlinear oscillators

Czechoslovak Journal of Physics, 2002

A pair of coupled classical oscillators with a general potential and general form of coupling is investigated. For general potentials, the single-frequency solution is shown to be stable for small excitations. For special potentials, such system remains stable for an arbitrary excitation. In both cases, the stability does not depend on the form of coupling. Transition to the instability regime

Stability analysis of two coupled oscillators

Mathematics and Mechanics of Complex Systems, 2016

We study a system of two coupled oscillators linked by a linear elastic spring and positioned vertically in a uniform gravity field. It is demonstrated that the system has different equilibrium configurations below and above the oscillators' suspension centers. We obtained the relations of the string stiffness and the distance between the suspension centers identifying the stability region of the oscillators.

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.

A degenerate bifurcation structure in the dynamics of coupled oscillators with essential stiffness nonlinearities

2003

We study the degenerate bifurcations of the nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to a nonlinear one that possesses essential stiffness nonlinearity. By defining the small coupling parameter ε, we study the dynamics of this system at the limit ε → 0. The degeneracy in the dynamics is manifested by a 'bifurcation from infinity' where a bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Another (nondegenerate) bifurcation point is generated close to the point of exact 1:1 internal resonance between the linear and nonlinear oscillators. The degenerate bifurcation structure can be directly attributed to the high degeneracy of the uncoupled system in the limit ε → 0, whose linearized structure possesses a double zero, and a conjugate pair of purely imaginary eigenvalues. First we construct local analytical approximations to the NNMs in the neighborhoods of the bifurcation points and at other energy ranges of the system. Then, we 'connect' the local approximations by global approximants, and identify global branches of NNMs where unstable and stable mode and inverse mode localization between the linear and nonlinear oscillators take place for decreasing energy.

Hybrid dynamics of two coupled oscillators that can impact a fixed stop

International Journal of Non-Linear Mechanics, 2003

We consider two linearly coupled masses, where one mass can have inelastic impacts with a fixed, rigid stop. This leads to the study of a two degree of freedom, piecewise linear, frictionless, unforced, constrained mechanical system. The system is governed by three types of dynamics: coupled harmonic oscillation, simple harmonic motion and discrete rebounds. Energy is dissipated discontinuously in discrete amounts, through impacts with the stop. We prove the existence of a nonzero measure set of orbits that lead to infinite impacts with the stop in a finite time. We show how to modify the mathematical model so that forward existence and uniqueness of solutions for all time is guaranteed. Existence of hybrid periodic orbits is shown. A geometrical interpretation of the dynamics based on action coordinates is used to visualize numerical simulation results for the asymptotic dynamics.

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.

A Model System for the Behavior of Two Non-Linearly Coupled Oscillators

Journal of Sound and Vibration, 1998

This paper presents a systematic analysis approach for the study of two non-linearly coupled oscillators, with incommensurable fundamental frequencies. The asymptotic perturbation method is used to analyze a bifurcation problem of codimension two. It is found that the amplitude modulation equations are equal to normal forms equations available in the literature. Approximate analytic solutions for generic quadratic and cubic non-linearities can be constructed. The results obtained from a computer simulation based on a fifth order Runge-Kutta-Fehlberg scheme confirm the validity of the asymptotic perturbation method. The method is illustrated by applying it to a two-rod system subjected to aerodynamic excitation.

Dynamics of a periodically driven chain of coupled nonlinear oscillators

Journal of Zhejiang University-SCIENCE A, 2017

A 1D chain of coupled oscillators is considered, including the Duffing-type nonlinearity, viscous damping, and kinematic harmonic excitation. The equations of motion are presented in a non-dimensional form. The approximate equations for the vibrational amplitudes and phases are derived by means of the classical averaging method. A simple analysis of the resulting equations allows one to determine the conditions for the two basic synchronous steady-states of the system: the in-phase and anti-phase motions. The relations between the required excitation frequency and the natural frequencies of the abbreviated (linear) system are discussed. The validity of these predictions is examined by a series of numerical experiments. The effect of the model parameters on the rate of synchronization is analyzed. For the purpose of systematic numerical studies, the cross-correlation of time-series is used as a measure of the phase adjustment between particular oscillators. Finally, some essential issues that arise in case of the mechanical system with dry friction are indicated.

Analysis of Control Relevant Coupled Nonlinear Oscillatory Systems

European Journal of Control, 2008

The paper proposes and analyzes two prototype structures of coupled generalized van der Pol equations able to describe self-excitation of simultaneous oscillations with distinct frequencies. These structures are relevant for describing oscillations phenomena which may be encountered on systems subject to control. These structures are analyzed using the Krylov-Bogoliubov averaging method. This analysis allows to establish conditions for the occurrence of the various operation regimes. The usefulness of the results is illustrated by their application to the straightforward analysis of the properties of a combustion instability model.