LIMITING PHASE TRAJECTORIES AND TRANSIENT RESONANCE OSCILLATIONS IN 1 AND 2 DOF ASYMMETRIC NONLINEAR SYSTEMS

2011 ◽  
Vol 21 (10) ◽  
pp. 2919-2928
Author(s):  
E. L. MANEVITCH ◽  
L. I. MANEVITCH
Keyword(s):  
Energy Exchange ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Conservation Of Energy ◽  
Phase Trajectories ◽  
Weakly Coupled ◽  
Limiting Phase Trajectories ◽  
Nonlinear Normal ◽  
Nonstationary Oscillations ◽  
Resonance Oscillations

The concept of limiting phase trajectories (LPT) has been introduced by one of the authors to describe intensive energy exchange between weakly coupled oscillators or oscillatory chains. It turns out that LPT can be considered as an alternative to nonlinear normal modes (NNMs), which are characterized by conservation of energy. LPT (in the introduced coordinates) describes the vibroimpact-type process with saw-tooth amplitude and a discontinuous derivative. It was shown earlier that this concept could also be extended to systems with one degree of freedom (DoF). In this case energy exchange between the oscillator and the source of energy can occur. In this paper, we generalize the above results in several ways, namely: (1) a consideration of the asymmetry of elastic potential; (2) a detailed description of the origin of vibroimpact-type behavior and the transition from nonresonant nonstationary oscillations to resonant ones (3) a direct application of obtained results to transient vibrations in strongly asymmetric 2DoF systems.

scholarly journals Limiting Phase Trajectories and Resonance Energy Transfer in a System of Two Coupled Oscillators

2010 ◽  
Vol 2010 ◽  
pp. 1-24 ◽  
Author(s):  
L. I. Manevitch ◽  
A. S. Kovaleva ◽  
E. L. Manevitch
Keyword(s):  
Energy Transfer ◽  
Coupled Oscillators ◽  
Energy Exchange ◽  
Duffing Oscillator ◽  
Normal Modes ◽  
Resonance Energy ◽  
Harmonic Excitation ◽  
Conservation Of Energy ◽  
Phase Trajectories ◽  
Limiting Phase Trajectories

We study a problem of energy exchange in a system of two coupled oscillators subject to 1 : 1 resonance. Our results exploit the concept of limiting phase trajectories (LPTs). The LPT, associated with full energy transfer, is, in certain sense, an alternative to nonlinear normal modes characterized by conservation of energy. We consider two benchmark examples. As a first example, we construct an LPT and examine the convergence to stationary oscillations for a Duffing oscillator subjected to resonance harmonic excitation. As a second example, we treat resonance oscillations in a system of two nonlinearly coupled oscillators. We demonstrate the reduction of the equations of motion to an equation of a single oscillator. It is shown that the most intense energy exchange and beating arise when motion of the equivalent oscillator is close to an LPT. Damped beating and the convergence to rest in a system with dissipation are demonstrated.

Nonlinear Dynamics of a System of Coupled Oscillators With Essential Stiffness Nonlinearities

2003 ◽  
Author(s):  
Alexander F. Vakakis ◽  
Richard H. Rand
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Resonance Capture ◽  
Periodic Motions ◽  
Internal Resonances ◽  
Dynamical Mechanism ◽  
Elliptic Orbits ◽  
Weakly Coupled ◽  
Resonant Dynamics ◽  
Nonlinear Normal

We study the resonant dynamics of a two-degree-of-freedom system composed a linear oscillator weakly coupled to a strongly nonlinear one, with an essential (nonlinearizable) cubic stiffness nonlinearity. 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 (nonlinear 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 nonlinear energy pumping phenomena from the linear to the nonlinear 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 confirm the analytical predictions.

Nonlinear Modes in Systems With Strong Nonlinearities

1997 ◽  
Author(s):  
Melvin E. King
Keyword(s):  
Differential Equation ◽  
Normal Modes ◽  
Approximate Solutions ◽  
Discrete Systems ◽  
Nonlinear Normal Modes ◽  
Algebraic Equations ◽  
Conservation Of Energy ◽  
Singular Ordinary Differential Equation ◽  
Power Series Expansions ◽  
Nonlinear Normal

Abstract In this paper, a symbolic/numeric method is developed to compute nonlinear normal modes (NNMs) in conservative, two-degree-of-freedom (2-DoF) systems. Based upon the notion of NNMs, periodic motions are sought during which the two coordinates ‘vibrate-in-unison’. By parameterizing the response of one coordinate with respect to the response of the other (reference) coordinate and by imposing conservation of energy, we obtain a nonlinear, singular ordinary differential equation. Approximate solutions for these modal functions are obtained, for a given energy level, via truncated power-series expansions. The coefficients of the expansion, along with the maximum and minimum reference displacements, are then computed by (i) symbolically evaluating the singular differential equation at various (distinct) reference displacements, and then (ii) numerically solving the resulting set of nonlinear algebraic equations. Since the approximate solution inherently depends upon the order of the expansion, convergence studies must be performed in order to ensure sufficient accuracy. Note that even though the formulation presented herein is based on 2-DoF systems, the methodology is quite general and can readily be extended to higher-order discrete systems. Moreover, since it does not rely upon any ‘small-quantity’ assumptions, it can be used to investigate the dynamics of coupled, strongly nonlinear systems.

Degenerate Bifurcation Scenarios in the Dynamics of Coupled Oscillators With Symmetric Nonlinearities

2005 ◽  
Author(s):  
O. V. Gendelman
Keyword(s):  
Coupling Parameter ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Saddle Node Bifurcation ◽  
Saddle Node ◽  
Weakly Coupled ◽  
Nonnegative Coefficients ◽  
Nonlinear Normal ◽  
Infinite Energy ◽  
Good Agreement

We study the degenerate bifurcations of the nonlinear normal modes (NNMs) of an unforced system consisting of a linear oscillator weakly coupled to an essentially nonlinear one. Both the potential of the oscillator and of the coupling spring are adopted to be even-power polynomials with nonnegative coefficients. By defining the coupling parameter ε, the dynamics of this system at the limit ε → 0 and for finite ε is investigated. Bifurcation scenario of the nonlinear normal modes is revealed. The degeneracy in the dynamics is manifested by a ‘bifurcation from infinity’ where a saddle-node bifurcation point is generated at high energies, as perturbation of a state of infinite energy. Another (nondegenerate) saddle-node bifurcation points (at least one point) are generated in the vicinity of the point of exact 1:1 internal resonance between the linear and nonlinear oscillators. The above bifurcations form multiple-branch structure with few stable and unstable branches. This structure may disappear (for certain choices of the oscillator and coupling potentials) by mechanism of successive cusp catastrophes with growth of the coupling parameter ε. The above analytical findings are verified by means of direct numerical simulation (conservative Poincare sections). For particular case of pure cubic nonlinearity of the oscillator and the coupling spring good agreement between quantitative analytical predictions and numerical results is observed.

An Energy-Based Formulation for Computing Nonlinear Normal Modes in Undamped Continuous Systems

1994 ◽  
Vol 116 (3) ◽  
pp. 332-340 ◽  
Author(s):  
M. E. King ◽  
A. F. Vakakis
Keyword(s):  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Conservation Of Energy ◽  
Nonlinear Modes ◽  
One Dimensional ◽  
Linear Differential ◽  
Nonlinear Normal ◽  
The Stability ◽  
Nonlinear Elastic

The nonlinear normal modes of a class of one-dimensional, conservative, continuous systems are examined. These are free, periodic motions during which all particles of the system reach their extremum amplitudes at the same instant of time. During a nonlinear normal mode, the motion of an arbitrary particle of the system is expressed in terms of the motion of a certain reference point by means of a modal function. Conservation of energy is imposed to construct a partial differential equation satisfied by the modal function, which is asymptotically solved using a perturbation methodology. The stability of the detected nonlinear modes is then investigated by expanding the corresponding variational equations in bases of orthogonal polynomials and analyzing the resulting set of linear differential equations with periodic coefficients by Floquet analysis. Applications of the general theory are given by computing the nonlinear normal modes of a simply-supported beam lying on a nonlinear elastic foundation, and of a cantilever beam possessing geometric nonlinearities.

Some Observations of Mode Localizations in Flexibly Coupled Two Rotors

2005 ◽  
Author(s):  
Yohta Kunitoh ◽  
Hiroshi Yabuno ◽  
Tsuyoshi Inoue ◽  
Yukio Ishida
Keyword(s):  
Multiple Scales ◽  
Method Of Multiple Scales ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Rotor System ◽  
The Other ◽  
Whirling Motion ◽  
Averaged Equations ◽  
Weakly Coupled ◽  
Nonlinear Normal

Mode localizations in a weakly coupled two-span rotor system are theoretically and experimentally discussed. One rotor has a slight unbalance and the other one is well-assembled. First, the equations governing the whirling motions of the coupled rotors are expressed due to nonlinearity in each span and the weakness of the coupling. The averaged equations are obtained by the method of multiple scales and it is shown that the nonlinear normal modes are bifurcated from the linear normal modes. It results from this bifurcation that the number of nonlinear normal modes exceeds the equivalent degree of freedom of the two-span rotor system, i.e., 2-degree under the assumption that the trajectory of the whirling motion is circle. Also, it is theoretically clarified that whirling motion caused by the unbalance in the rotor is localized in the rotor with unbalance or in one without unbalance depending on the rotational speed. Furthermore, these mode localizations are experimentally confirmed.

Localized and Non-Localized Nonlinear Normal Modes in a Multi-Span Beam With Geometric Nonlinearities

1996 ◽  
Vol 118 (4) ◽  
pp. 533-542 ◽  
Author(s):  
J. Aubrecht ◽  
A. F. Vakakis
Keyword(s):  
Coupling Parameter ◽  
Galerkin Approximation ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Finite Amplitude ◽  
Torsional Stiffness ◽  
Mode Localization ◽  
Weakly Nonlinear ◽  
Weakly Coupled ◽  
Nonlinear Normal

The nonlinear normal modes of a geometrically nonlinear multi-span beam consisting of n segments, coupled by means of torsional stiffeners are examined. Assuming that the stiffeners possess large torsional stiffness, the beam displacements are decomposed into static and flexible components. It is shown that the static components are much smaller in magnitude than the flexible ones. A Galerkin approximation is subsequently employed to discretize the problem, whereby the computation of the nonlinear normal modes of the multi-span beam is reduced to the study of the periodic solutions of a set of weakly coupled, weakly nonlinear ordinary differential equations. Numerous stable and unstable, localized and non-localized nonlinear normal modes of the multi-span beam are detected. Assemblies consisting of n = 2, 3, and 4 beam segments are examined, and are found to possess stable, strongly localized nonlinear normal modes. These are free synchronous oscillations during which only one segment of the assembly vibrates with finite amplitude. As the number of periodic segments increases, the structure of the nonlinear normal modes becomes increasingly more complicated. In the multi-span beams examined, nonlinear mode localization is generated through two distinct mechanisms: through Pitchfork or Saddle-node mode bifurcations, or as the limit of a continuous mode branch when a coupling parameter tends to zero.

Nonlinear Normal Modes in a Class of Nonlinear Continuous Systems

1993 ◽  
Author(s):  
Melvin E. King ◽  
Alexander F. Vakakis
Keyword(s):  
Reference Point ◽  
Invariant Manifolds ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Continuous Systems ◽  
Partial Differential ◽  
Analytical Methodology ◽  
Conservation Of Energy ◽  
Linearized Stability ◽  
Nonlinear Normal

Abstract A general methodology is developed for computing the nonlinear normal modes of a class of undamped vibratory systems governed by nonlinear partial differential equations of motion. A nonlinear normal mode is defined as free motion during which all points of the system vibrate equiperiodically, reaching their extremum positions at the same instants of time. The analytical methodology is based on a previous work by Shaw and Pierre (1992b), where the displacements and velocities at any point of a structure were expressed as functions of the displacement and velocity of a single reference point. The dynamics of the continuous system were then restricted to invariant manifolds of the phase space. Motivated by the methodology presented by Shaw and Pierre, we express the displacement of an arbitrary point of the structure as a function of the displacement of a single reference point. Assuming undamped oscillations (and thus conservation of energy), a singular partial differential equation for the function relating the displacements is derived, and is subsequently solved using an asymptotic, power series methodology. Applications of the general theory are then given by computing the nonlinear normal modes of a simply supported beam resting on a nonlinear elastic foundation, and of a cantilever beam having geometric nonlinearities. The stability of the detected modes is then investigated by a linearized stability analysis.

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